The Republican billionaire's victory on November 8 saw him clinch the crucial Electoral College count, which determines the presidency, but lose the popular vote to rival Hillary Clinton by more than two million ballots.
Cloistered in his Florida resort for the long Thanksgiving weekend, the 70-year-old tycoon who has never previously held elected office took to Twitter to indulge in one of his customary tweet storms.
On Sunday, before returning by private jet to New York to resume interviews with potential cabinet appointees, he claimed he would have won the popular vote if it were not for "the millions of people who voted illegally."
"Serious voter fraud in Virginia, New Hampshire and California -- so why isn't the media reporting on this? Serious bias -- big problem!" he followed up later.
Trump, who spent the campaign warning that the result might be "rigged," is now -- with his aides -- pushing back hard as the Green Party works to secure recounts in three states which Trump won: Michigan, Pennsylvania and Wisconsin.
The campaign of Green Party candidate Jill Stein, who won a tiny fraction of the vote in each state and where Trump beat Clinton by thousands of votes, said voters had filed recount requests in more than 100 (out of 9,163) precincts in Pennsylvania -- promising that more would follow.
The party requested a statewide recount in Wisconsin on Friday and plans to request a recount in Michigan on Wednesday, the campaign said.
- 'No evidence' -
But observers deny any evidence of widespread fraud.
And few expect any change in the outcome of the vote, which former secretary of state Clinton conceded to Trump in an early-hours phone call on November 9.
Meanwhile, Trump increased his Electoral College vote count to 306, compared to 232 for Clinton, after Michigan certified its election results -- and his victory there.
Republican, Democratic and independent lawmakers dismissed Trump's claims as totally unsubstantiated.
Some experts warned they set a dangerous precedent by potentially undermining trust in democracy or confidence in his leadership.
"I have not seen anything in the millions, I don't know what he was talking about," Republican Senator James Lankford told CNN.
White House spokesman Josh Earnest said "there has been no evidence produced to substantiate a claim like that."
Clinton's campaign has said it would join the process, but has also indicated it so far sees no evidence of hacking or vote tampering.
Bernie Sanders, who ran against Clinton for the Democratic nomination, called Trump's remarks "unfounded nonsense" that showed Republicans wanted to make it harder for people including minorities to register to vote.
- Discord -
In New Hampshire, deputy secretary of state David Scanlan told AFP that isolated instances of voter fraud "show up in every election" but that the 2016 ballot had run "very smoothly."
But the dispute roils what has already been a rough transition period, as Trump continues to hold back-to-back meetings with people he is considering for cabinet posts.
Trump has picked Georgia Representative Tom Price, a robust critic of President Barack Obama's health care reform law, to serve as health secretary, and plans to make a formal announcement as early as Tuesday, US media reported.
And his Vice president-elect Mike Pence told reporters to expect "a number of very important announcements" on Tuesday, fueling speculation of more cabinet posts to be revealed.
Trump threatened to end the thaw in US relations with Cuba, following the death of Fidel Castro, unless Havana makes concessions on human rights and opening up its economy.
Discord has flared in the Trump camp over the pick for secretary of state, the most prestigious job in the cabinet -- in particular over the candidacy of Mitt Romney, the 2012 Republican presidential nominee.
Other potential candidates are outspoken former New York mayor Rudy Giuliani; David Petraeus, the celebrated general who later resigned as CIA director and pleaded guilty to showing classified material to his mistress; and Bob Corker, a Republican lawmaker and chairman of the Senate Foreign Relations Committee.
Trump met Petraeus on Monday, and was to meet Romney for a second time on Tuesday, in addition to a planned meeting with Corker.
"Very impressed!" tweeted Trump of Petraeus just minutes after the general left the building, having described their conversation as "very good."
Romney was one of Trump's most outspoken critics during the campaign, and Trump aide Kellyanne Conway declared, in an unusual statement by a senior political aide, that she had received "a deluge" of concern from supporters about him.
