President Trump pushed on Tuesday to hand the executive branch unprecedented influence over how federal elections are run, signing a far-reaching and legally dubious order to change U.S. voting rules.

The executive order, which seeks to require proof of citizenship to register to vote as well as the return of all mail ballots by Election Day, is an attempt to upend centuries of settled election law and federal-state relations.

The Constitution gives the president no explicit authority to regulate elections. Instead, it gives states the power to set the “times, places and manner” of elections, leaving them to decide the rules, oversee voting and try to prevent fraud. Congress can also pass election laws or override state legislation, as it did with the Voting Rights Act of 1965.

Yet Mr. Trump’s order, which follows a yearslong Republican push to tighten voting laws out of a false belief that the 2020 election was rigged, bypasses both the states and Congress. Republican lawmakers in Washington are trying to pass many of the same voting restrictions, but they are unlikely to make it through the Senate.

The order’s most eye-catching provisions are the requirements of proof of citizenship and the return of mail ballots by Election Day.

But the order, which threatens to withhold federal funding from states that do not comply with it, includes a range of other measures.

It seeks to give federal agencies, including the Elon Musk-led team known as the Department of Government Efficiency, access to state voter rolls to check “for consistency with federal requirements.” It aims to set new rules for election equipment, which could force states to replace voting machines that use bar codes or QR codes. And it instructs the U.S. attorney general to hunt for and prosecute election crimes.

Is this legal?

Probably not all of it, legal experts say — and voting rights groups and state attorneys general are already signaling that they will file challenges.

Several experts predicted that provisions of the order might well be found unlawful, though they said that others, like directions to Mr. Trump’s attorney general and other cabinet members, fell within legal bounds.

“It’s an attempt at a power grab,” said Richard L. Hasen, an election law expert at the University of California, Los Angeles. “The president has been seen in the past as having no role to play when it comes to the conduct of federal elections, and this attempt to assert authority over the conduct of federal elections would take power away from both an independent federal agency and from the states.”

What parts might be challenged?

A central question surrounds Mr. Trump’s attempt to use the Election Assistance Commission, a federal agency that Congress created in 2002 to help election officials with their work, to enforce the proof-of-citizenship requirement.

Currently, Americans may register to vote in federal elections either through their state or by using a federal form created by the E.A.C. The form includes a box that registrants check to attest, under penalty of perjury, that they are U.S. citizens, but it does not require documentation as proof.

The executive order would force the E.A.C. to change that process to require a passport, state identification that includes citizenship information or military identification.

Legal experts dispute that Mr. Trump has the authority to force the agency, which Congress designated as “independent” and which includes two commissioners from each party, to take any action.

“He can ask nicely,” said Justin Levitt, a professor of constitutional law at Loyola Marymount University who served in the Biden administration. “But he thinks he’s got a power that, at least so far, he does not have. It would take a change in the law and the Supreme Court affirmatively approving a radical expansion of power of the executive.”

Legal experts say the provision requiring all ballots to arrive by Election Day also probably exceeds the president’s legal authority, particularly the threat to withhold federal funding from those states that do not comply. (Seventeen states currently allow mail ballots postmarked by Election Day to be counted if they arrive soon afterward.)

“If the president is basically usurping the power of the purse by imposing limits on these grants that Congress itself did not impose, that could be the basis for constitutionally challenging these conditions,” said Laurence H. Tribe, a professor emeritus at Harvard Law School.

Mr. Trump’s attempt to force states to turn over voter data to Mr. Musk’s team and federal agencies recalls a similar program from the first Trump administration, a commission on “election integrity” led by Kris Kobach, who is now the Kansas attorney general.

The commission sought data from all 50 states, but 44 of them refused to comply. The Republican secretary of state in Mississippi told the commission to “go jump in the Gulf of Mexico.”

What is the potential impact on voters and elections?

If the full order were to stand, it could potentially disenfranchise millions of Americans and cost state and local governments millions of dollars.

About 21.3 million people do not have proof of citizenship readily available, according to a 2023 study by the Brennan Center for Justice, a voting rights and democracy group. Nearly four million people do not have the documents at all because they were lost, destroyed or stolen. The executive order does not allow for birth certificates to prove citizenship.

It is also unclear whether women who have changed their surname after marriage will face new hurdles in proving their citizenship.

The order could also lead election officials to throw out sizable numbers of ballots that arrive after Election Day. For example, in Nevada’s two largest counties in the 2022 general election, about 45,000 ballots arrived after Election Day and were counted, according to state data. In Washington State, Kim Wyman, a Republican former secretary of state, estimated that “about a third of the ballots in any given election” arrived on the Wednesday or Thursday after Election Day.

The order could put states in deep budget holes, as well. Many states, including the battlegrounds of Georgia and Pennsylvania, use voting machines with bar codes or QR codes. Replacing them would cost millions of dollars that the order does not provide.

Republicans won in 2024. Why do they want to change voting laws?

Mr. Trump has made specious claims about voter fraud for decades, but since his 2020 election loss and the 2021 Capitol riot, he has pushed the issue to the center of Republican politics.

Even though voter fraud is exceedingly rare, nearly every speech of Mr. Trump’s 2024 campaign included false claims that noncitizens were voting in American elections. He also railed against mail voting, even as Republican groups successfully pushed more of the party’s voters to cast ballots that way.

Jason Snead, the executive director of the Honest Elections Project, a conservative advocacy group tied to the activist Leonard Leo, said the executive order was simply enforcing laws already passed by Congress. He referred to a ruling from a federal appeals court that found that Congress’s selection of a federal Election Day meant all voting must be completed by that day, with no late-arriving ballots permitted.

“The executive order is acting well within the four corners of those existing laws, so we’re not breaking new ground in terms of legal authority,” Mr. Snead said. “We’re not breaking new ground in terms of the relationship between the federal government and the states.”