Former New Jersey Governor Chris Christie called the president's legal team a "national embarrassment".
President Trump has refused to concede the election, making unsubstantiated claims of widespread electoral fraud.
Many Republicans have supported his legal efforts but a small but growing number have broken ranks.
On Saturday Mr Trump suffered a major blow in Pennsylvania, after a judge dismissed a lawsuit from the Trump campaign seeking to invalidate millions of mail-in votes in the battleground state.
In a scathing ruling, Judge Matthew Brann said his court had been presented with "strained legal arguments without merit and speculative accusations".
The move paves the way for Pennsylvania to on Monday certify Mr Biden's win - the Democratic president-elect leads there by more than 80,000 votes.
Mr Biden is projected to defeat President Trump 306 to 232 in the US electoral college, which determines who becomes president - far above the 270 he needs to win.
Speaking to ABC's This Week programme on Sunday, Mr Christie, a former New Jersey governor, said: "Quite frankly, the conduct of the president's legal team has been a national embarrassment".
He said the Trump camp was often discussing election fraud "outside the courtroom, but when they go inside the courtroom they don't plead fraud and they don't argue fraud".
"I have been a supporter of the president's. I voted for him twice. But elections have consequences, and we cannot continue to act as if something happened here that didn't happen."
Mr Christie was the first governor to endorse Mr Trump as a presidential candidate back in 2016. He also helped prepare the US president for his debates with Mr Biden earlier this year.
He singled out for criticism Sidney Powell, a lawyer who appeared with Trump's legal team during a press conference on Thursday who, without providing evidence, said that electronic voting systems switched millions of ballots to Biden, and that he also won thanks to "communist money".
But on Sunday the Trump campaign issued a statement distancing themselves from Ms Powell, saying she "is practicing law on her own" and was "not a member of the Trump legal team".
A tweet from President Trump earlier this month explicitly named her as part of the team.
On Sunday other Republicans also urged President Trump to concede.
Maryland Governor Larry Hogan told CNN on Sunday that the Trump camp's continued efforts to overturn the election results were "beginning to look like we're a banana republic".
In a post on Twitter, Gov Hogan said Mr Trump should "stop golfing and concede".
Michigan Representative Fred Upton told CNN the voters in his battleground state "spoke" by choosing Mr Biden, while North Dakota Senator Kevin Cramer said in an interview to NBC it was "past time to start a transition", although he stopped short of accepting Mr Biden's victory.
President Trump's refusal to concede has complicated the process that normally follows a US election.
Joe Biden is to announce his first cabinet appointments on Tuesday, said Ron Klain, the president-elect's choice as White House chief-of-staff.
Speaking to ABC on Sunday, Mr Klain called on the Trump administration to facilitate the transition.
"A record number of Americans rejected the Trump presidency, and since then Donald Trump's been rejecting democracy," he said.
And he revealed that the Biden campaign was preparing for a "scaled-down" inauguration on 20 January due to the worsening coronavirus outbreak.