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.
“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”.