President Biden and former President Donald Trump on Wednesday agreed to hold two campaign debates — the first on June 27 hosted by CNN and the second on Sept. 10 hosted by ABC — setting the stage for their first presidential face-off to play out in just over a month.

The quick agreement on the timetable followed President Biden’s announcement that he would not participate in fall presidential debates sponsored by the nonpartisan commission that has organized them for more than three decades. This includes a debate scheduled for Virginia State University on Oct. 1.

“Virginia State University was excited to make history as the first HBCU to host a presidential debate,” Virginia Congresswoman Jennifer McCellan said in a statement. “I hope the ongoing negotiations between the two campaigns result in a presidential debate at VSU, and I have requested that the Biden-Harris campaign engage with Virginia State University students and the Tri-Cities community with an event before the election”

President Biden’s campaign proposed that media outlets directly organize the debates between the presumptive Democratic and Republican nominees.

The debate is so unusually early on the political calendar that neither President Biden nor Mr. Trump will have formally accepted his party’s nomination.

Hours later, President Biden said he had accepted an invitation from CNN, adding “Over to you, Donald.”

Mr. Trump, who had insisted he would debate President Biden anytime and anyplace, said on Truth Social he’d be there, too, adding, “Let’s get ready to Rumble!!!” Soon after that, they agreed to the second debate on ABC.

“Trump says he’ll arrange his own transportation,” President Biden wrote on X, working in a jab about the perks of incumbency. “I’ll bring my plane, too. I plan on keeping it for another four years.”

The swiftness with which the match-ups came together reflects how each of the two unpopular candidates thinks he can get the better of his opponent in a head-to-head showdown. Mr. Trump and his team are convinced the debates will exacerbate voters’

concerns about President Biden’s age and competence, while the Biden team believes the Trump often-incendiary rhetoric will remind voters of why they voted him out of the White House four years ago.

The presidential debates, always a critical moment on the political calendar, could be particularly important in a year when voters are underwhelmed with their choices and have expressed concerns about the candidates’ advanced ages — President Biden is 81 and Mr. Trump 77.

Sprightly on social media, the rivals traded barbs — each claiming victory the last time they faced off in 2020.

“Donald Trump lost two debates to me in 2020, since then he hasn’t shown up for a debate,” President Biden said in a post on X. “Now he’s acting like he wants to debate me again. Well, make my day, pal.”

Mr. Trump, for his part, said the president was the “WORST debater I have ever faced - He can’t put two sentences together!”

The June debate is likely to cap a busy and unsettled stretch, following the likely conclusion of Mr. Trump’s criminal hush money trial in New York, foreign trips by President Biden to France and Italy, the end of the Supreme Court’s term, and the expected start of two criminal trials for the president’s son, Hunter Biden.

CNN said that its debate would be held at 9 p.m. ET in its Atlanta studios with no audience present in a break from recent precedent. Moderators will be anchors Jake Tapper and Dana Bash, CNN said. ABC did not offer details on where its event would be held. Disagreements about moderators and rules were some of the questions that prompted the formation of the Commission on Presidential Debates in 1987.

The two campaigns and television networks had held weeks of informal talks on ways to circumvent the commission’s grip on presidential debates following years of complaints and perceived slights, according to two people familiar with the matter who spoke about internal discussions on condition of anonymity.

The Biden campaign had proposed excluding third-party candidates, such as Robert F. Kennedy Jr., from the debates outright. Under the debate commission’s rules, Mr. Kennedy or other third-party candidates could qualify if they secured ballot access sufficient to claim 270 Electoral Votes and polled at 15% or higher in a selection of national surveys.

Both CNN and ABC announced the same qualification threshold, saying candidates will need to reach at least 15% in four separate national polls of registered or likely voters that meet their standards.

In response, Mr. Kennedy accused President Biden and Mr. Trump of “trying to exclude me from their debate because they are afraid I would win.” He said, “Keeping viable candidates off the debate stage undermines democracy.”

The commission said in a Wednesday statement, “The American public deserves substantive debates from the leading candidates for president and vice president.” It said its mission is “to ensure that such debates reliably take place and reach the widest television, radio and streaming audience.”