WASHINGTON—President Donald Trump has selected a nominee for the U.S. Supreme Court and will introduce him to a national television audience Tuesday evening, a crucial early moment for his administration that could shape American law for decades.

Mr. Trump has chosen either Judge Thomas Hardiman of the Third U.S. Circuit Court of Appeals in Philadelphia or Judge Neil Gorsuch of the 10th U.S. Circuit Court of Appeals in Denver, a person familiar with the selection process said Monday.

With the aid of senior advisers such as White House Counsel Don McGahn, the president last week narrowed the list to a handful of federal appellate judges admired in conservative circles, four of whom he interviewed personally, people familiar with the process said.

Mr. Trump said on Twitter early Monday he had made his decision. Speaking later to small-business owners in the White House, the president didn’t reveal the nominee’s identity but said the selection is “a person who is unbelievably highly respected.…I think you will be very impressed with this person.”

The announcement is likely to trigger a tussle in Congress, between the White House and its critics, and among a host of interest groups as they battle over a nomination that would restore a conservative majority to the evenly split court, replacing the influential Justice Antonin Scalia, who was revered by conservative scholars and activists.