If confirmed, the judge will have a lifetime appointment and therefore may be at the heart of the American political process long after Mr Trump has left office.
Whoever is nominated is sure to face bitter opposition from Democrats, still infuriated at the Republican obstruction that kept Barack Obama from filling the year-long vacancy on the court.
The next justice will also play a key role as the court considers abortion access, gay and transgender rights and legal challenges to key aspects of Mr Trump’s agenda.
Here are the leading candidates:
Home state: Colorado
Current position: Federal judge on 10th circuit court of appeals
If the rumblings in Washington are to be believed, Mr Gorsuch is the front-runner to be elevated to America’s highest court.
His relative youth makes him a particularly intriguing selection for Mr Trump, as he could well serve for decades. If confirmed, he would be the youngest justice to join the court since William Rehnquist in 1971.
A Harvard Law School alumnus with a doctorate in legal philosophy from Oxford, he clerked for two Supreme Court justices and is considered a gifted writer of legal opinions.
Mr Gorsuch is an avowed originalist, meaning he believes Constitution is a stable rather than living document, and has sided with Christian groups in religious freedom cases.
William Pryor Jr
Home state: Alabama
Current position: Federal judge on 11th circuit court of appeals
Mr Pryor met with Mr Trump earlier this month, establishing him as a leading candidate.
He is perhaps the most outwardly political of those on the short list, and Senate Democrats tried to block him from reaching the federal bench in 2004.
Mr Pryor is staunchly anti-abortion, having called the Roe v Wade that legalised abortion the "worst abomination of constitutional law in our history”.
He served as Alabama’s attorney general from 1997-2004, and is an ally of Jeff Sessions, the Alabama senator nominated by Mr Trump to head the justice department.
In 2003, Mr Pryor helped force Alabama’s chief justice out of office for refusing to remove a Ten Commandments monument from the state Supreme Court.
Home state: Pennsylvania
Current position: Federal judge on 3rd circuit court of appeals
Mr Hardiman is a guns-rights advocate and has consistently ruled in favour of police powers.
Unlike Mr Pryor he won unanimous senate confirmation for a federal judgeship in 2007, though he has since written a few controversial opinions, including one allowing strip-searches for all new prisoners at a county jail.
He was educated at Notre Dame and Georgetown universities and worked at private law firms in Pittsburgh and Washington before joining the bench.
Mr Hardiman has not received as much attention as Mr Gorsuch and Mr Pryor in the run-up to Mr Trump’s announcement but reportedly joins them on the president's short list.