The top contenders also included federal appeals judges Raymond Kethledge, Amy Coney Barrett and Thomas Hardiman.
While the president pondered his choice, his aides prepared for what is expected to be a tough confirmation fight. Judge Brett Kavanaugh looked to have the inside track within the White House last week but may have hit a snag. Jon Kyl of Arizona would guide the nominee through the Senate confirmation process. He works for the Washington-based lobbying firm Covington & Burling. The White House hoped to keep the details under wraps until he rolls out his pick from the East Room.
Kennedy was long a swing vote on the nine-member court, and Trump's choice - his second opportunity in 18 months to fill a Supreme Court vacancy - will dramatically affect many aspects of American life, from abortion to voting rights to immigration.
Hardiman was the runner-up when Trump nominated Neil Gorsuch to replace the late Justice Antonin Scalia previous year.
The president has a plethora of excellent candidates and Republicans could support any one of them, the Utah Republican said on Fox News. But the situation appeared to remain fluid.
Judge Thomas Hardiman is a US appeals court judge that The Washington Post reports is a "Second Amendment extremist". And Hardiman, the first in his family to graduate from college, has the kind of personal story that appeals to many Trump supporters.
Kavanaugh, 53, earned his law degree from Yale University and was also an editor on the prestigious Yale Law Journal. Kennedy sometimes joined the liberal justices on key rulings on divisive social issues like abortion and gay rights, a practice his replacement may not duplicate.
Mike Pompeo dismisses North Korea's 'gangster' comments, says talks are going well
He also acknowledged that the road ahead would be hard , but said that North Korea did reaffirm its commitment to denuclearize. And he said some progress had been made towards agreeing "the modalities" of North Korea's destruction of a missile facility.
Seen as a dark horse to replace retiring Justice Anthony Kennedy, conservative groups, senators and opinion leaders made a case for Lee over the weekend.
Barrett - a longtime Notre Dame Law School professor who became a federal judge last fall - excited social conservatives since she was questioned about her Roman Catholic faith in her nomination hearings a year ago, but her brief time on the bench has raised questions about her experience.
Barret, the only woman on Trump's shortlist, was appointed past year to the 7th U.S. Circuit Court of Appeals in Chicago. A self-described introvert, Kethledge co-authored a book, "Lead Yourself First", in which he talks about how some of the world's great leaders learned from solitude and quiet.
He cited the "third-party doctrine" established in 1979, which allows law enforcement to obtain data on a person without a warrant if they obtained it from a third party.
Kennedy, 81, announced on June 27 plans to retire after three decades on the court, effective on July 31. Without Republican defections, however, Senate rules leave Democrats with scant options to block confirmation of Trump's nominee.
Still, Senator Roy Blunt of Missouri, a member of the Republican leadership, said on NBC's "Meet the Press" on Sunday that the Senate "can confirm any of the four names being mentioned". I think they'd be fine justices of the Supreme Court.
"I will oppose the nomination the President will make tonight because it represents a corrupt bargain with the far Right, big corporations, and Washington special interests", Casey said in a statement on his website. "And I expect we will do that on sort of a normal timetable, a couple of months".