Mr. Trump has peppered associates with questions about Judge Hardiman. Some close to the president said he found Judge Hardiman’s personal story, offered to him by the judge’s supporters, to be compelling. Judge Hardiman was the first member of his family to graduate from college, and he helped pay for his education by driving a taxi.

Judge Hardiman has also had an important supporter within the Trump family. He served with Judge Maryanne Trump Barry, Mr. Trump’s sister, on the United States Court of Appeals for the Third Circuit, in Philadelphia. Judge Barry recommended Judge Hardiman to her brother last year as a good choice for the court, according to two people close to Mr. Trump.

Judge Hardiman’s appointment would also bring some educational diversity to a Supreme Court awash in Ivy League diplomas. Judge Hardiman attended the University of Notre Dame and Georgetown University Law Center.

But Mr. Trump has also said positive things to associates about Judge Amy Coney Barrett, a staunch social conservative, the people familiar with the process said. He spent part of Sunday at his club with the Fox News host Sean Hannity, who favors Judge Barrett, people familiar with their meeting said.

Mr. Trump also continued to discuss Judge Brett M. Kavanaugh, a former staff secretary to George W. Bush, who was a favorite of the White House counsel, Donald F. McGahn II, and who had been seen as the favorite last week.

The only judge among the four whom the president appeared to lose much of his interest in was Raymond M. Kethledge. People close to the process said the president had found him likable but comparatively dull. And some conservatives, whose support has guided Mr. Trump’s thinking about the courts, have voiced concern about Judge Kethledge on issues like immigration, with some comparing him to Justice David H. Souter, who was appointed by the first President George Bush and who disappointed some Republicans with his votes.

Still, Mr. Trump was said to be struggling to get past Judge Kavanaugh’s connection to the Bush family. Mr. Trump savaged Jeb Bush during the 2016 primary, and the president has remained suspicious about the Bushes.