Senate Majority Leader Mitch McConnell and House Minority Leader Kevin McCarthy both dismissed the idea, BBC reported on Friday.
Senator McConnell said no US presidential election had ever been delayed before.
"Never in the history of this country, through wars, depressions and the Civil War, have we ever not had a federally scheduled election on time. We will find a way to do that again this November third," he said.
McCarthy echoed him. "Never in the history of the federal elections have we ever not held an election and we should go forward with our election," he said.
Trump ally Senator Lindsay Graham meanwhile said a delay was "not a good idea".
US House Speaker Nancy Pelosi from Trump’s rival Democratic Party responded to his tweet floating the idea of moving the November election by tweeting a quote from the Constitution that cites the congressional authority needed to do so.
Trump appears to be doing everything in his power to undermine the credibility of November's vote, in which a record number of Americans are predicted to rely on mail-in voting to avoid the risk of exposure to the coronavirus. He's repeatedly made false and misleading claims about the reliability of the mail balloting and suggested broad conspiracy theories. Critics warn that he could be laying the groundwork for contesting the results – although the purpose may be simply to give him a scapegoat if he loses.
At a Thursday afternoon White House news conference, Trump denied he wanted to postpone the election, but argued that mass postal voting would leave the result in doubt.
"I don't want to delay, I want to have the election," he said. "But I also don't want to have to wait for three months and then find out that the ballots are all missing and the election doesn't mean anything."
In a series of tweets earlier, Trump railed against mass postal voting and warned – without providing evidence – that it would be susceptible to foreign interference.
President Trump does not have the authority to move the date of the election, which by law is held on the first Tuesday after the first Monday in November.
Any change of date would need to be approved by both houses of Congress – the House of Representatives and the Senate. Democrats control the House of Representatives and some have already said they will not support any delay to the vote.
Any move by Congress to delay the election into 2021 would also require a constitutional amendment, US media quoted constitutional experts as saying.