The sweeping order also would seek to take federal funding away from states that do not comply.
Trump has long questioned the US electoral system and continues to falsely claim that his 2020 loss to Democratic President Joe Biden was the result of widespread fraud.
The president and his Republican allies also have made baseless claims about widespread voting by non-citizens, which is illegal and rarely occurs.
Last year the Republican-controlled House of Representatives approved a bill that would ban non-citizens from registering to vote in federal elections, a practice that is already illegal. It did not pass the Senate, which was then controlled by Democrats.
The White House's order seeks to achieve similar goals. Voting rights groups argued that it, like the Safeguard American Voter Eligibility Act that did not become law, would disenfranchise voters, particularly people of colour, who do not have access to passports or other required identification.
"We've got to straighten out our elections," Trump said on Tuesday as he signed the order at the White House.
"This country is so sick because of the elections, the fake elections and the bad elections, we're going to straighten that out one way or the other."
The order is likely to draw legal challenges.
"This is a blatant attack on democracy and an authoritarian power grab," said Lisa Gilbert, co-president of advocacy group Public Citizen.
In recent years Republicans have tried to put more restrictions on voting, while Democrats have sought to make it easier to vote by supporting mail-in ballot access and early voting opportunities.
Public Citizen noted about 146 million Americans do not have a passport, and Brennan Center research showed 9 per cent of US citizens who are eligible to vote, or 21.3 million people, do not have proof of citizenship "readily available".
According to US law, the US secretary of state can unilaterally cancel passports if it determines they were "illegally, fraudulently, or erroneously obtained" or created through illegality or fraud.
The White House argued that Trump's order would prevent foreign nationals from interfering in US elections. Under the new directive, voters would be asked a citizenship question on the federal voting form for the first time.
"Federal election-related funds will be conditioned on states complying with the integrity measures set forth by federal law, including the requirement that states use the national mail voter registration form that will now require proof of citizenship," a White House fact sheet about the order said.
The order criticised policies allowing mail-in ballots to arrive and be counted after election day. The order said Trump's policy is to "require that votes be cast and received by the election date established in law."
