Trump is considering several candidates to replace Ruth Bader Ginsburg, the liberal justice who died Friday at age 87.
Two federal appeals court judges, Amy Coney Barrett of the 7th Circuit, and Barbara Lagoa of the 11th Circuit, are considered front-runners for the nomination.
The Republican president said in a Twitter post he would make his announcement of his pick at the White House, at a time to be determined.
If Trump follows through on that timing, his selection will be revealed a day after Ginsburg lies in state at the U.S. Capitol, the first woman to be granted that honor.
Sen. Mitt Romney, R-Utah, said Tuesday he will vote on Trump’s pick if the nomination reaches the Senate floor.
Romney’s announcement makes it much more likely that Trump’s nominee will get a vote in the Senate. Republicans can afford only three defections to approve Trump’ pick in the Senate, where they hold a 53-47 lead.
Before Romney committed to a vote, two other GOP senators, Lisa Murkowski of Alaska and Maine’s Susan Collins, had said they opposed voting on the nomination before the presidential election in November.
Democrats oppose Trump naming a nominee, at least until the presidential election is resolved.
Democrats argue that if Joe Biden wins the election, he should be allowed to pick Ginsburg’s replacement, not a lame duck Trump.
But in a statement, Romney said: “The Constitution gives the President the power to nominate and the Senate the authority to provide advice and consent on Supreme Court nominees.”
“Accordingly, I intend to follow the Constitution and precedent in considering the President’s nominee. If the nominee reaches the Senate floor, I intend to vote based upon their qualifications,” Romney said.
However, 52% of voters in six so-called swing states want the winner of the election to name the next high court justice, according to a CNBC/Change Research poll released Tuesday.
And just 37% of voters nationally think Trump should be able to nominate a justice if he loses, according to the poll. Fifty-seven percent said he should not be able to do so.