Republican leader Mitch McConnell has insisted there will be an “orderly” post-election transition - something the president has cast doubt on.
Mr McConnell said regardless of who wins the 3 November presidential election, there will be a peaceful inauguration on 20 January.
A day earlier, President Donald Trump refused to commit to this, saying “we’ll have to see what happens”.
Mr Trump has questioned the integrity of US postal voting in recent months.
The president currently trails his challenger, Democrat Joe Biden, in national opinion polls with 40 days to go until the election.
As many more Americans will be casting their votes by post this year due to the pandemic, Mr Trump has been questioning the security of this mail-in ballot system.
Every losing presidential candidate in modern times has conceded. If Mr Trump were to refuse to accept the result of the election, it would take the country into uncharted territory.
Mr Biden has suggested that should this happen, the military could remove Mr Trump from the White House.
What have Republicans said?
“The winner of the November 3rd election will be inaugurated on January 20th,” Mr McConnell tweeted on Thursday.
“There will be an orderly transition just as there has been every four years since 1792.”
Other Republican lawmakers, including vocal Trump ally Senator Lindsey Graham, have similarly promised a safe and fair election.
“I can assure it will be peaceful,” Mr Graham told Fox News.
Senator Mitt Romney offered a more critical response on Wednesday, saying “any suggestion that a president might not respect this Constitutional guarantee is both unthinkable and unacceptable”.
What did Trump say?
Mr Trump was asked by a reporter on Wednesday evening if he would commit to a peaceful transfer of power “win, lose or draw” to Mr Biden.
“I’ve been complaining very strongly about the ballots,” Mr Trump said. “And the ballots are a disaster.”
When the journalist countered that “people are rioting”, Mr Trump interjected: “Get rid of the ballots, and you’ll have a very - you’ll have a very peaceful - there won’t be a transfer, frankly, there’ll be a continuation.”
Back in 2016, Mr Trump also refused to commit to accepting the election results in his contest against the Democratic candidate, Hillary Clinton, which she characterised as an attack on democracy.
He was eventually declared the winner, although he lost the popular vote by three million, an outcome he still questioned.
What have Democrats said?
Democratic Speaker of the House Nancy Pelosi, the third most powerful politician in Washington, told reporters on Thursday that she was not surprised at Mr Trump’s earlier remarks.
Mrs Pelosi added that the president “admires people who are perpetuating their role in government”, citing Russia’s Vladimir Putin, North Korea’s Kim Jong-un and Turkey’s Recep Tayyip Erdogan.
“But I remind him: you are not in North Korea, you are not in Turkey, you are not in Russia, Mr President… so why don’t you just try for a moment to honour your oath of office.”
Speaking to reporters in Delaware, Mr Biden said Mr Trump’s comments on the transition of power were “irrational”.
The Democrat’s team also said “the United States government is perfectly capable of escorting trespassers out of the White House”.
Mr Biden has himself been accused by conservatives of stoking unrest over the election by saying in August: “Does anyone believe there will be less violence in America if Donald Trump is re-elected?”
Last month, Mrs Clinton urged Mr Biden to not concede defeat early on election night. “I think this is going to drag out, and eventually I do believe he will win if we don’t give an inch.”
She raised the scenario that Republicans would try “messing up absentee balloting” and mobilise an army of lawyers to contest the result.
Doubts about the fairness of November’s vote come as another high-stakes political battle is fought - on whether or not to appoint a new Supreme Court justice before the election.