President Biden is willing to debate former President Donald J. Trump at least twice before the election, and as early as June — but his campaign is rejecting the nonpartisan organization that has managed presidential debates since 1988, according to a letter obtained by The New York Times.

The letter by the Biden campaign lays out for the first time the president’s terms for giving Mr. Trump what he has openly clamored for: a televised confrontation with a successor Mr. Trump has portrayed, and hopes to reveal, as too feeble to hold the job. In a Truth Social post on Wednesday morning, Mr. Trump quickly agreed to the two dates proposed by the Biden campaign, although it was unclear whether he would agree to Mr. Biden’s other terms.

Mr. Biden and his top aides want the debates to start much sooner than the dates proposed by the Commission on Presidential Debates, so voters can see the two candidates side by side well before early voting begins in September. They want the debate to occur inside a TV studio, with microphones that automatically cut off when a speaker’s time limit elapses. And they want it to be just the two candidates and the moderator — without the raucous in-person audiences that Mr. Trump feeds on and without the participation of Robert F. Kennedy Jr. or other independent or third-party candidates.

The proposal suggests that Mr. Biden is willing to take some calculated risks to reverse his fortunes in a race in which most battleground-state polls show the president trailing Mr. Trump and struggling to persuade voters that he’s an effective leader and steward of the economy.

It is the first formal offer by the Biden campaign for debates with Mr. Trump, who has declared repeatedly that he will debate his successor “anytime and anywhere,” and has demanded as many debates as possible. Mr. Biden recently indicated he would debate Mr. Trump, but had until now declined to give any firm commitment or specific details.

The letter, signed by Mr. Biden’s campaign chair, Jennifer O’Malley Dillon, and addressed to the Commission on Presidential Debates, notifies the group that Mr. Biden will not be participating in the three general-election debates sponsored by the commission, which are scheduled for Sept. 16, Oct. 1 and Oct. 9.

It is a striking decision for Mr. Biden, an institutionalist who has tried to preserve the traditions of Washington.

Instead, Ms. O’Malley Dillon writes in the letter that Mr. Biden will participate in debates hosted by news organizations. The move opens the doors for the Biden team and potentially the Trump team to negotiate directly with networks — and with one another — for possible debates.

In a video announcing his offer, Mr. Biden taunted Mr. Trump. “Make my day, pal,” he said, adding a reference to the one weekday Mr. Trump’s Manhattan trial is generally not in session. “Let’s pick the date, Donald. I hear you’re free on Wednesdays.

Mr. Trump, in his insult-laden response, said he would like to see more than two debates and for “excitement purposes, a very large venue.” Calling Mr. Biden “the WORST” debater and “crooked,” he accused the president of being “afraid of crowds.”

Ms. O’Malley Dillon suggested that the first debate be held in late June, by which time Mr. Trump’s New York criminal trial should be completed and after Mr. Biden returns from the Group of 7 summit meetings with other heads of state.

A second presidential debate should be held “in early September at the start of the fall campaign season, early enough to influence early voting, but not so late as to require the candidates to leave the campaign trail in the critical late September and October period,” she writes.

The Biden campaign also proposes that one vice-presidential debate be held in late July after Mr. Trump and his running mate are formally nominated at the Republican National Convention in Milwaukee.

For the president, early debates hold significant advantages. Early votes are crucial, especially for Democrats. And polls show that Mr. Biden currently trails Mr. Trump and that his messages on core issues like the economy are not resonating with enough voters.

In the 2020 election, Democrats put a huge emphasis on voting early by mail as a safe alternative to in-person voting during the coronavirus pandemic. Early votes gave Mr. Biden a decisive edge over Mr. Trump, who had told his voters not to trust the mail and to instead vote only on Election Day.

Mr. Trump and the Republican National Committee have tried to repair that damage this year by telling Republicans to vote early.

“The commission’s failure, yet again, to schedule debates that will be meaningful to all voters — not just those who cast their ballots late in the fall or on Election Day — underscores the serious limitations of its outdated approach,” Ms. O’Malley Dillon writes in the letter.

Mr. Trump leads Mr. Biden in most polls of battleground states, including the recent surveys by The New York Times, Siena College and The Philadelphia Inquirer. Significantly more voters trust Mr. Trump over Mr. Biden to handle the economy.

The Biden campaign and the president’s White House staff widely feel that the debates were important in 2020, and that they will be important again this year.

The Biden campaign has been trying to remind voters of why a majority removed Mr. Trump from office in 2020. People close to the president have said they’re worried about so-called Trump amnesia — that voters are nostalgic about Mr. Trump and have forgotten how divisive he was — and some of the recent polling underscores that point.

A side-by-side debate, which could have a large viewing audience, is the most dramatic way for the Biden campaign to give Mr. Trump more exposure, in their view.

In the first debate in 2020, Mr. Trump barely allowed Mr. Biden to get a word in. He was aggressive and constantly interrupting, while sweating and appearing unwell. Mr. Biden, exasperated, famously said to Mr. Trump, “Will you shut up, man? This is so unpresidential.” And in the days following that first debate, Mr. Trump’s poll numbers fell.

The Trump campaign’s top officials, Susie Wiles and Chris LaCivita, see the situation differently and share their boss’s eagerness for him to debate Mr. Biden as often as possible. They have indicated that they don’t care who hosts the debate, or where it’s held. The Trump campaign believes, almost to a person, that Mr. Biden has declined significantly since 2020 and would be exposed in a debate against Mr. Trump.

The letter from Ms. O’Malley Dillon could spell the end of a storied organization that has been running presidential debates since the Reagan era. She makes clear to the commission in her letter that the Biden campaign does not trust the organization to conduct a professional debate, saying it “was unable or unwilling to enforce the rules in the 2020 debates.”

Among other grievances with the commission, Biden aides are still furious that Mr. Trump debated Mr. Biden in 2020 and appeared visibly under the weather, announcing soon after the debate that he had tested positive for the coronavirus. The Biden team was also livid that members of the Trump family took their masks off when they arrived in the audience for the debate.

Still, the Biden campaign’s debate proposal comes with conditions. And the decision to sideline the commission offers clear advantages to Mr. Biden. For starters, the Biden campaign proposes limiting the number of debates to only two, whereas the commission has already scheduled three presidential debates.

Biden campaign officials want the debates to be held in a television studio without an in-person audience that could cheer, boo and derail the conversation, as Trump supporters did during a CNN town hall last year. The commission always invites an audience to watch its presidential debates.

There’s also a chance that Mr. Kennedy reaches the 15 percent national polling threshold to qualify for the commission’s debates. The Biden campaign views Mr. Kennedy as a spoiler candidate and people close to the president worry that with the Kennedy name he could attract support from voters who might otherwise support Mr. Biden.

Ms. O’Malley Dillon writes in her letter that the debate should be one-on-one to allow voters “to compare the only two candidates with any statistical chance of prevailing in the Electoral College — and not squandering debate time on candidates with no prospect of becoming president.”

The Biden campaign has proposed rules — including the automatic cutting-off of microphones — to ensure Mr. Trump does not blow through his time limits and talk over Mr. Biden as he did relentlessly during their first debate in 2020.

“There should be firm time limits for answers, and alternate turns to speak — so that the time is evenly divided and we have an exchange of views, not a spectacle of mutual interruption,” Ms. O’Malley Dillon writes in the letter.

“A candidate’s microphone should only be active when it is his turn to speak, to promote adherence to the rules and orderly proceedings.”

The Biden campaign has also proposed criteria to limit which television networks are allowed to host the debate. It should only be hosted, Ms. O’Malley Dillon writes, by broadcast organizations that hosted both a Republican primary debate in 2016 in which Mr. Trump participated and a Democratic primary debate in 2020 in which Mr. Biden participated — “so neither campaign can assert that the sponsoring organization is obviously unacceptable.”

Networks that meet that mark include CBS News, ABC News, CNN and Telemundo.

And the debate moderators “should be selected by the broadcast host from among their regular personnel, so as to avoid a ‘ringer’ or partisan,” Ms. O’Malley Dillon adds.

The absence of an audience could be a sticking point for Mr. Trump, who has often played to crowds at debates and in town halls, encouraged by their applause, catcalls and jeers.

Nonetheless, the Trump campaign has been complaining about the commission for months.

In a statement on May 1 condemning the organization, Ms. Wiles and Mr. LaCivita blasted the group for not agreeing to earlier debates given the fact that early voting begins well before Election Day.

“We must host debates earlier than ever before,” they said. “Again, we call on every television network in America that wishes to host a debate to extend an invitation to our campaign and we will gladly negotiate with the Biden campaign, with or without the stubborn Presidential Debates Commission.”

For decades, candidates in both parties have criticized the commission. In 2000, George W. Bush’s campaign tried to engineer its own schedule of debates, but ultimately consented to debates led by the organization.

In 2012, Republicans complained bitterly about the debates between Mitt Romney, their nominee, and the incumbent, President Barack Obama, when a moderator fact-checked Mr. Romney in real time during one debate.

In 2016, the Trump campaign fought with the commission over the seating of four women in the Trump family’s box at a debate, three of whom had accused Hillary Clinton’s husband, former President Bill Clinton, of sexual misconduct.

And in 2020, both the Trump and Biden teams struggled with the commission. Mr. Trump boycotted the second scheduled debate, which the organization decided to make a virtual event.

In 2022, the Republican National Committee — which has no direct role in negotiating presidential debates with the commission — voted unanimously to have the party nominee pull out of debates with the organization.

Reid J. Epstein contributed reporting.