CNN’s White House correspondent, Kaitlan Collins, said her sources had told her the President’s son-in-law, who is married to Ivanka Trump, and senior adviser has raised the question of concession with him.
His campaign issued a statement saying that Mr Biden is “rushing to falsely pose as the winner” and that the race is “far from over”.
“I will not rest until the American People have the honest vote count they deserve and that Democracy demands,” Mr Trump said in the statement, adding that the campaign’s legal battle will begin Monday.
Mr Trump also continued to claim that he was true election winner, taking to Twitter to repeat earlier allegations that observers were not allowed into ballot counting rooms. Writing in capital letters, he said: “I won the election, got 71,000,000 legal votes. Bad things happened which our observers were not allowed to see.”
The Biden campaign earlier said there had been no communication between the president-elect and Mr Trump since the race was called.
Mr Trump has filed a raft of lawsuits to challenge the results, but elections officials in states across the country say there has been no evidence of significant fraud. Legal experts say Mr Trump’s efforts are unlikely to succeed.
Judges have already thrown out cases in Georgia, Michigan and Nevada. In Pennsylvania, judges sided with Republicans and ordered some provisional ballots set aside and granted Republican observers greater access to vote counting.
Experts have said the legal challenges were too narrow in scope to have an impact on the outcome of the election.
Meanwhile, Republicans are trying to raise at least $60 million to fund legal challenges, sources told Reuters.
“He should make sure every vote is counted and demand transparency. That puts him on solid rhetorical grounds,” said another former White House official.
Republicans are worried that Mr Trump could tarnish his legacy if he does not eventually make a graceful exit, eroding his future political power. “It will be impossible for him to run again in 2024 if he’s seen as a sore loser,” a congressional Republican source said.
Fox News host Laura Ingraham, a staunch Trump defender, on Friday urged the president, if and when the time came, to accept an unfavourable outcome with “grace and composure,” and the Wall Street Journal’s conservative editorial board wrote that Mr Trump “needs evidence to prove voter fraud.”
One Trump adviser said White House Chief of Staff Mark Meadows would likely be the aide who would raise the idea of conceding with the president. Mr Meadows came down with the coronavirus this week and is under quarantine.