A number of Republican leaders have condemned US President Donald Trump's claims of "major fraud" in the election, and have rebuked his efforts to halt "all voting" counts while declaring himself the election victor.

Trump, who spoke to supporters at the White House late on election night, when millions of votes were still uncounted, said: "As far as I'm concerned, we already have won it."

"This is a fraud on the American public. This is an embarrassment to our country. We were getting ready to win this election. Frankly, we did win this election. We did win this election," he said.

But several GOP leaders have since publicly shot down the comments.

Republican Senate Majority Leader Mitch McConnell said it was "not unusual" for people to claim they had won the election.

"But claiming you win the election is different from finishing the (vote) counting," he said.

Former New Jersey Governor Chris Christie, a top ally of Trump, told ABC News he disagreed with the President's remarks.

"I talk tonight now not as a former governor, but as a former US attorney," Christie said on election night.

"There's just no basis to make that argument tonight. There just isn't.

"All these votes have to be counted that are in now. In Pennsylvania, the argument won't even start in Pennsylvania until tomorrow, or Thursday or Friday because the Pennsylvania court extended for three days when you could accept ballots.

"I understand that there could be an argument on that, based on Pennsylvania law, but that argument is for later. Tonight, this was not the time to make this argument."

Christie added: "I disagree with what he did tonight."

"There comes a point where you have to let the process play itself out before you judge it to have been flawed.

"I think it's a bad strategic decision, it's a bad political decision and it's not the kind of decision you would expect someone to make tonight who holds the position he holds," Christie said.

Fellow Trump ally, former Pennsylvania Senator Rick Santorum, told the CNN election coverage panel he was "very distressed" by what the President had said.

"The idea of using the word 'fraud' being committed by people counting votes is wrong," Santorum said.

"(In Pennsylvania) they're counting the absentee and mail-in ballots right now. And some counties have stopped counting.

"Why have they stopped counting? Because it's 2.48 in the morning!

"That's why the stopped counting. People get tired, when they're tired they make mistakes, and you stop counting and then you take it up the next morning when people are fresh.

"They're not stopping counting because they're trying to 'fix' anything to create any sort of fraud."

He said in regards to President Trump's comments: "I could not disagree more in this case."

Rob Portman, a Republican senator from Ohio, issued a statement with an indirect rebuke of the President's rhetoric.

"Under our Constitution, state legislatures set the rules and states administer our elections," Portman said.

"We should respect that process and ensure that all ballots cast in accordance with state laws are counted. It's that simple."

Trump's former national security adviser John Bolton, who has become a public critic, said they were "some of the most irresponsible comments that a president of the United States has ever made".

"He has cast doubt on the integrity of the entire electoral process purely for his own personal advantage," he told Sky News.

"It's a disgrace."

American conservative political commentator Ben Shapiro also joined in, disagreeing with Trump's decision to declare victory prematurely.

A winner in the election has yet to be declared, but the Trump campaign has begun mounting numerous legal fights to stop the counting of votes in key battleground states and has already pledged to seek a recount in the state of Wisconsin.