Jay Sekulow, the chief counsel for the American Center for Law and Justice, said that fierce Democratic opposition to any nominee could prompt Republicans, who control the Senate, to change the rules to require a simple majority vote to confirm a court justice. Democrats did that for lower-court judges when they were in control, and Republicans could return the favor.

But Mr. Sekulow predicted that Mr. Trump’s nominee would be confirmed in the end, no matter what.

“The conservative groups will mobilize, are already mobilized,” he said. “This is Justice Scalia’s seat. Conservatives cannot afford for that seat to be lost.”

It is possible that Mr. Trump will pick another nominee from his list of 21 possibilities, besides Judges Pryor, Gorsuch or Hardiman — or someone else entirely. Among the judges who have also received serious consideration are Diane S. Sykes, 59, of the federal appeals court in Chicago. But Judge Sykes’s chances seem to be fading, perhaps because her age is on the high side by the standards of recent nominees. All of Mr. Trump’s candidates have conservative records that are certain to become the focus of intense scrutiny and debate.

Judge Hardiman, 51, has an appealing personal story to tell, based partly on a humble background that distinguishes him from most of the current justices. He was the first person in his family to attend college, at Notre Dame, and he drove a cab to pay for law school, at Georgetown.

In 2003, President George W. Bush appointed him to the Federal District Court in Pittsburgh. Only one member of the current Supreme Court, Justice Sonia Sotomayor, has such trial court experience. Mr. Bush elevated Judge Hardiman in 2007 to the United States Court of Appeals for the Third Circuit, where he served alongside Mr. Trump’s sister, Judge Maryanne Trump Barry.

But Judge Gorsuch and Judge Pryor appear to be the most likely choices, and both have conservative records that are certain to become the focus of intense scrutiny and debate.