Judge Thomas M. Hardiman, the first member of his family to graduate from college, has a personal story President Trump is said to like.

BERKELEY HEIGHTS, N.J. — President Trump sought to mine a last bit of drama from his decision on a Supreme Court nominee on Sunday, saying he might need to extend the process well into Monday, just hours before he is scheduled to announce the pick in a prime-time address.

“I’m very close to making a decision,” Mr. Trump said on Sunday afternoon as he boarded Air Force One to return to Washington after a weekend spent golfing at his private club in Bedminster, N.J., and soliciting opinions from dozens of people about what he should do.

Mr. Trump added that he has “not made it final,” calling all four of his final choices to replace Justice Anthony M. Kennedy “excellent,” and adding, “You can’t go wrong.”

Yet Mr. Trump is mindful that he has little room for error, given Republicans’ razor-thin majority in the Senate and the looming midterm elections, and he would prefer a choice who can be confirmed relatively easily, according to people who have spoken to him.

He gave himself an extra 12 hours to make his choice, telling reporters it would come before noon on Monday. He had previously said the choice would be made by Sunday, before his announcement of the nomination at 9 p.m. Monday in a nationally televised event.

Mr. Trump, while an incorrigible gossip, can be effective at keeping a secret when he chooses to, and he loves the drama surrounding an important nomination. Over the past three days, he has stoked uncertainty among even his closest aides by asking lots of questions but offering little in return, according to those who have spoken with him.

The suspense-laden process bore similarities, but also some differences, to how the president handled the announcement last year of his first Supreme Court nominee, Neil M. Gorsuch. White House officials worked hard to encourage speculation over the pick until just before Judge Gorsuch was announced in another prime-time address.

But in that case, the president notified Judge Gorsuch that he was the choice a day ahead of the announcement, and Mr. Trump was said to have settled on the judge days before that.

The runner-up last year was Judge Thomas M. Hardiman. While a late addition to Mr. Trump’s final four this time around, in recent days the president has expressed fresh interest in him, according to people close to the selection process.

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…