Last week, Trump had tweeted that he would be announcing the pick this coming Thursday, which also happens to be Groundhog Day.
While on the campaign trail, Trump released a list of 11 names of people he might consider to fill the vacancy left by the death of Justice Antonin Scalia nearly a year ago.
The three leading contenders are thought to be federal appeals court judges William Pryor, Thomas Hardiman, and Neil Gorsuch, sources familiar with the process told NBC News. Those judges were nominated to their current posts by President George W. Bush.
Senate Minority Leader Chuck Schumer has previously pointed to Republican inaction on President Obama's Supreme Court nominee, Merrick Garland, as an example of what could happen to a Trump pick.
"We are not going to make it easy for them to pick a Supreme Court Justice," Schumer, D-N.Y., told Rachel Maddow on MSNBC earlier this month. "It's hard for me imagine a nominee that Donald Trump would choose that would get Republican support that we would support."