Trump floated a delay until people could "properly, securely and safely" vote.
There is little evidence to support Mr Trump's claims but he has long railed against mail-in voting which he has said would be susceptible to fraud.
US states want to make postal voting easier due to public health concerns over the coronavirus pandemic.
Under the US constitution, Mr Trump does not have the authority to postpone the election himself. Any delay would have to be approved by Congress. The president does not have direct power over the two houses of Congress.
In a series of tweets, Mr Trump said "universal mail-in voting" would make November's vote the "most inaccurate and fraudulent election in history" and a "great embarrassment to the USA".
He suggested - without providing evidence - that mail-in voting, as it is known in the US, would be susceptible to foreign interference.
Mr Trump also said postal voting was "already proving to be a catastrophic disaster" in areas where it was being tried out.
In June, New York allowed voters to vote by post in the Democratic primary poll for the party's presidential candidate. But there have been long delays in counting the ballots and the results are still unknown.
US media report that there are also concerns that many ballots will not be counted because they were not filled in correctly or do not have postmarks on them that show they were sent before voting officially ended.
However, several other states have long conducted votes by post.
Donald Trump can't delay November's presidential election without Congress, partially controlled by the Democrats, first approving the decision. If he didn't already know this, someone has certainly told him by now.
The president also must know that tweeting about a delay - even framed as an "I'm just asking!" question - is sure to ignite a political firestorm, particularly after he has repeatedly refused to say whether he'd accept an adverse result in the upcoming presidential election.
Mr 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.
His tweet could also be an attempt to divert attention away from the truly dismal second-quarter economic numbers just released. He's been relying on a financial turnaround to breathe life into his re-election campaign, and instead the outlook appears exceedingly gloomy.
Whatever the reason, tweeting about an election delay is not the move of a candidate confident of victory - and could be a sign of more desperate moves to come.
Quizzed by reporters on whether a president could delay the election, Secretary of State Mike Pompeo said he would not "enter a legal judgement on the fly". When pressed, he said the justice department would "make that legal determination", adding "we want an election that everyone is confident in".
Numerous Republicans - including Senate Majority Leader Mitch McConnell and House of Representatives Minority Leader Kevin McCarthy - also dismissed the idea.
Democrats have also been lining up to condemn Mr Trump's suggestion. New Mexico Senator Tom Udall said there was "no way" the president could delay the election.
However Chris Stewart, a Republican congressman from Utah, said that while he did not support delaying the election, Mr Trump had a legitimate point about postal voting being hard to monitor.
President Trump does not have the authority to move the date of the election, which is usually held on the first Tuesday in November.
Any change of date would need to be approved by the two 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. The amendment would be needed to change the dates for swearing in members of Congress and the new presidential administration, according to NPR.
Finally, legal experts said that even if Congress did agree to delay the election, Mr Trump's own term as president would still expire by 20 January 2021 under the constitution's 20th Amendment.
Earlier this month, six US states were planning to hold "all-mail" ballot elections in November: California, Utah, Hawaii, Colorado, Oregon and Washington. Other states are considering it, according to a postal voting campaign group.
These states will automatically send postal ballots to all registered voters, which then have to be sent back or dropped off on election day - although some in-person voting is still available in certain limited circumstances.
About half of US states allow any registered voter to cast their ballot by post on request.
Donald Trump suggests delay to 2020 US presidential election