- The pair traded barbs on social media, with Mr Trump stating: "Just tell me when. Let's get ready to rumble!!!"
- But the president's proposals break several traditions established over decades of presidential campaign.
Joe Biden has proposed two televised US presidential election debates, which rival Donald Trump quickly accepted.
The first is proposed for 27 June and the second would be in September.
The pair traded barbs on social media, with Mr Trump stating: "Just tell me when. Let's get ready to rumble!!!"
Mr Biden said his opponent was "free on Wednesdays" - a reference to Mr Trump's criminal trial, while Mr Trump claimed his rival was the "worst debater" who "can't put two sentences together".
But the president's proposals break several traditions established over decades of presidential campaign.
The debates would bypass the presidential debate commission, which since 1988 has scheduled the timing of these election-year faceoffs and had already set dates and locations for three debates in the autumn.
According to US media, Mr Biden's team wants two televised debates to be conducted with no live audience - another break with tradition.
Candidates would take turns answering question from an agreed-upon moderator from a major news network, and the non-speaking candidate's microphone would be deactivated.
Jen O'Malley Dillon, the Biden campaign chairwoman, said said that raucous partisan crowds were not "conducive to good debates". The first 2020 debate between the two candidates was marred by frequent interruptions and cross-talk.
The pair traded barbs on social media after Mr Biden announced his proposal on Wednesday.
"Donald Trump lost two debates to me in 2020. Since then he hasn't shown up for a debate," Biden said in a video message posted on Twitter/X.
"Now he's acting like he wants to debate me again. Make my day, pal. I'll even do it twice."
On TruthSocial, Mr Trump replied: "I am Ready and Willing to Debate Crooked Joe at the two proposed times in June and September.
"I would strongly recommend more than two debates and, for excitement purposes, a very large venue, although Biden is supposedly afraid of crowds - that's only because he doesn't get them," Trump added.
No third candidate?
If the two campaigns bypass the presidential debate commission, they would be able to decide whether any third-party candidates - including Robert F Kennedy Jr - could participate. In the past, the nonpartisan commission has set polling and ballot-access requirements for inclusion.
Mr Kennedy has yet to meet those qualifications, although he is polling better than any third-party candidate since Ross Perot in 1996, amid widespread voter dissatisfaction with both Mr Trump and Mr Biden.
Polling is mixed about which of the major-party candidates he damages more.
By moving the debates earlier in the presidential campaign calendar, Mr Biden's proposal could also lower any impact the debates have on the outcome of the November presidential election, giving both candidates more time to recover from a poor performance.
Some past incumbent presidents - including Mr Trump, Barack Obama, George HW Bush and Ronald Regan - have had shaky showings in their first general election debate before regaining their footing in subsequent meetings.
A June debate would take place before both the Republican and Democratic national conventions, while the September debate would be at least a month before election day. Surveys indicate many Americans don't begin paying attention to US presidential campaign news until the autumn.
Mr Biden also proposed a July vice-presidential debate that would take place after Mr Trump's running-mate was nominated at the Republican convention.
Rusty debating skills
The election-year presidential debates are typically broadcast on multiple US networks. At least 73 million Americans watched the first 2020 debate between Mr Trump and Mr Biden.
Mr Trump declined to participate in any Republican primary debates during this campaign, a point noted by Mr Biden in the video his campaign released announcing the debate proposal.
"Donald Trump lost two debates to me in 2020," Biden said. "Since then, he hasn't shown up for a debate. Now he's acting like he wants to debate me again."
Mr Biden had minimal opposition in his bid for nomination this year, and the Democratic Party held no primary debates.