Multiple congressional Republicans — including Senate Majority Leader Mitch McConnell and top House of Representatives Republican Kevin McCarthy — rejected the idea.
"Never in the history of the federal elections have we ever not held an election and we should go forward with our election," said McCarthy.
Democratic US Representative Zoe Lofgren, who chairs the House committee overseeing election security, also rejected any delay.
"Under no circumstances will we consider doing so to accommodate the President's inept and haphazard response to the coronavirus pandemic, or give credence to the lies and misinformation he spreads," Lofgren said in an email to Reuters.
Trump wrote on Twitter later on Thursday that Americans needed to know the election results on the night of the voting, not days or months later. Mail-in ballots can sometimes take longer to count.
"Glad I was able to get the very dishonest LameStream Media to finally start talking about the RISKS to our Democracy from dangerous Universal Mail-In-Voting," he said in the tweet.
Trump's tweets will deepen Democrats' fears that he will try to interfere with the election or refuse to accept its outcome should he lose.
Joe Biden has called that his biggest fear and has gone so far as to suggest sending in the military to force Trump out if he refused to leave.
Democrats are preparing for fights over absentee ballots, potential voting recounts and the possibility that Trump's Republican supporters will seek to intimidate voters at the polls.
The truly dangerous part of Trump's tweet on Thursday was not his suggestion of delaying the election — which is a "fantasy".
But his latest claim that voting by mail is rife with fraud, according to Justin Levitt, a professor at Loyola Marymount University and a constitutional law expert.
"This is yet another example of the president seeking to delegitimize the election process before it happens," Levitt said. "That is deeply destabilizing."
A Republican close to the White House was stunned at the tweet, noting that it followed a period of stability in which Trump has stayed on message in response to advice from new campaign manager Bill Stepien and senior campaign manager Jason Miller.