The Republican president’s campaign accused organisers of this week’s showdown of helping the Democrat by leaving out foreign policy as a topic, BBC reported.
The Biden camp shot back that Mr Trump was trying to avoid questions about his response to the coronavirus pandemic.
Mr Biden has a commanding lead nationally in opinion polls with two weeks to go until the election.
But he has a smaller lead in the handful of key US states that will ultimately decide the outcome.
What did the Trump campaign say?
On Monday, the president’s camp sent a letter to the Commission on Presidential Debates calling for topics to be adjusted for the final primetime duel this Thursday.
Trump campaign manager Bill Stepien said in the letter that the campaigns had already agreed foreign policy would be the focus of the third debate.
The topics were announced by moderator and NBC News correspondent Kristen Welker last week: American families, race in America, climate change, national security and leadership.
During a campaign rally on Monday afternoon in Prescott, Arizona, Mr Trump described Ms Welker as a “radical Democrat” and said she would be “no good”.
Mr Stepien accused Mr Biden of being “desperate to avoid conversations about his own foreign policy record” and the commission of trying to “insulate Biden from his own history”.
“The Commission’s pro-Biden antics have turned the entire debate season into a fiasco and it is little wonder why the public has lost faith in its objectivity,” he wrote.
He also accused Mr Biden of trying to avoid questions over reports about purported emails from his son, Hunter, and alleged conflicts of interest.
How did the Biden campaign respond?
The Democrat’s camp hit back that it was actually Mr Trump who was trying to duck questions.
“The campaigns and the Commission agreed months ago that the debate moderator would choose the topics,” said national press secretary TJ Ducklo.
“The Trump campaign is lying about that now because Donald Trump is afraid to face more questions about his disastrous Covid response.
“As usual, the president is more concerned with the rules of a debate than he is getting a nation in crisis the help it needs.”
What are the debate rules?
Following public criticism over the handling of the first debate, the commission has adopted a new rule to mute microphones in the final event.
The 90-minute debate structure will be divided into 15-minute segments. At the start of each new topic, both candidates will have two minutes of uninterrupted time – during which the opponent’s microphone will be off.
The rest of the time will be open discussion – and the microphones will not be muted during this period.
In a statement announcing the decision, the debate commission said they determined it was “appropriate to adopt measures intended to promote adherence to agree upon rules”.
The commission noted that “one [campaign] may think they go too far, and one may think they do not go far enough”, but that these actions provided the right balance in the interests of the public.
What happened with the last two debates?
The Trump campaign chief noted on Monday that the moderator of the cancelled second debate on 15 October, Steve Scully, had been suspended after tweeting to a prominent Trump critic, then lying that his account had been hacked.
Mr Stepien also accused the moderator of the first debate, Fox News’ Chris Wallace, of having acted as “a third combatant” against Mr Trump.
The first Trump-Biden duel back on 29 September descended into insults between the candidates, with the president interrupting many more times than the Democrat did, according to post-debate statistics from US media outlets.
How is early voting going?
Nearly 30 million early voters have already cast their ballots, compared with just six million at this point before the last presidential election in 2016.
Experts say the coronavirus pandemic has spurred many to cast their ballot ahead of time to avoid crowding at polling stations on 3 November, though some early voters have faced long queues.
On Monday, Republicans were dealt a defeat by the US Supreme Court as it declined to take up a case on postal ballots in the critical swing-voting state of Pennsylvania.
Republicans had argued only ballots received by Election Day should be counted, and were contesting a state Supreme Court decision to allow late ballots to count.
Now that America’s highest court has refused to hear the case, any ballots received within three days of 3 November will be counted, even if they do not have a clear postmark.
Chief Justice John Roberts sided with the court’s three liberal justices in the case.