Trump for months before the election made repeated claims without providing evidence that results would be marred by fraud and has kept up those unfounded allegations over the past week.
Judges have tossed out lawsuits in Michigan and Georgia, and experts say Trump's legal efforts have little chance of changing the election result.
But Congress's top Republican, Senate Majority Leader Mitch McConnell, on Monday lined up behind Trump, saying that he was "100% within his rights to look into allegations of irregularities," without citing any evidence.
The dispute is slowing Biden's work in preparing for the work of governing, as a Trump appointee who heads the office charged with recognizing election results has not yet done so.
Biden on Saturday secured the more than the 270 votes in the Electoral College needed to win the presidency. He also led Trump in the popular vote by 4.6 million votes on Tuesday morning as states continued to count the remaining ballots.
BARR MOVE PROMPTS RESIGNATION
Barr's directive to prosecutors prompted the top lawyer overseeing voter fraud investigations to resign in protest.
Barr told prosecutors on Monday that "fanciful or far-fetched claims" should not be a basis for investigation and his letter did not indicate the Justice Department had uncovered voting irregularities affecting the outcome of the election.
But he did say he was authorizing prosecutors to "pursue substantial allegations" of irregularities of voting and the counting of ballots.
Richard Pilger, who for years has served as director of the Election Crimes Branch, said in an internal email he was resigning from his post after he read "the new policy and its ramifications".
The previous Justice Department policy, designed to avoid interjecting the federal government into election campaigns, had discouraged overt investigations "until the election in question has been concluded, its results certified, and all recounts and election contests concluded."
Biden's campaign said Barr was fueling Trump's far-fetched allegations of fraud.
"Those are the very kind of claims that the president and his lawyers are making unsuccessfully every day, as their lawsuits are laughed out of one court after another," said Bob Bauer, a senior adviser to Biden.
One of Barr's predecessors as attorney general, Republican Alberto Gonzales, told CNN on Tuesday the timing of Barr's memo was "very, very unfortunate" because it contributes to the perception that the Justice Department was being used for political purposes.
"If you're asking me do I think there have I seen evidence of widespread fraud, to a level that would overturn the results of this election? No, I have not," said Gonzales, who served under former President George W. Bush.
Although a few Republicans have urged Trump to concede, the president still held the support of prominent party leaders who had yet to congratulate Biden.
Trump's campaign on Monday filed a lawsuit to block Pennsylvania officials from certifying Biden's victory in the battleground state, where Biden led by more than 45,000 votes.
It alleged the state's mail-in voting system violated the US Constitution by creating "an illegal two-tiered voting system" where voting in person was subject to more oversight than voting by mail.
It was filed against Pennsylvania Secretary of State Kathy Boockvar and the boards of elections in Democratic-leaning counties that include Philadelphia and Pittsburgh. Boockvar's office did not immediately respond to a request for comment.
"The Trump campaign's latest filing is another attempt to throw out legal votes," Pennsylvania Attorney General Josh Shapiro, a Democrat, said on Twitter.
Pennsylvania state Representative Dawn Keefer on Tuesday planned to call for a legislative audit of the state's election results.
Biden, who has begun work on his transition to the White House, will give a speech on Tuesday defending the Affordable Care Act, the landmark healthcare law popularly known as Obamacare, as the US Supreme Court hears arguments on a lawsuit backed by the Trump administration to invalidate it.
Trump and Republicans have repeatedly tried to do away with the 2010 law passed under President Barack Obama, with Biden as his vice president.
The Supreme Court fended off previous challenges in 2012 and 2015. The court now has a 6-3 conservative majority after Trump's third appointee, Amy Coney Barrett, was confirmed last month.
As Biden begins work on his transition, his team is considering legal action over a federal agency's delay in recognizing his victory over Trump.
The General Services Administration normally recognizes a presidential candidate when it becomes clear who has won so a transition of power can begin.
But that has not yet happened and the law does not spell out when the GSA must act. GSA Administrator Emily Murphy, appointed by Trump in 2017, has not yet determined that "a winner is clear," a spokeswoman said.
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