President Trump has said he will name a replacement to Ruth Bader Ginsburg by week’s end and urged the Republican-controlled Senate to confirm his Supreme Court choice before 3 November.
The plan has launched a high-stakes battle ahead of the election.
Mr Trump would replace Ginsburg, a liberal stalwart who died on Friday aged 87, with a conservative.
The president appears to have secured enough support in the US Senate to win approval for his nominee.
This would cement a right-leaning majority on the court for decades.
The ideological balance of the nine-member court is crucial to its rulings on the most important issues in US law.
On Monday Mr Trump said that he was “constitutionally obligated” to nominate someone for the Supreme Court.
The president earlier had a private meeting at the White House with a potential nominee: Amy Coney Barrett, an appeals court judge who is backed by anti-abortion conservatives.
Once the president names a nominee, it is the Senate’s job to vote on whether to confirm them. The Judiciary Committee will review the pick first, and then vote to send the nominee to the floor for a full vote.
Senate majority leader Mitch McConnell has vowed to hold a confirmation vote before the election in November. Democrats have accused him of hypocrisy.
Following the death of conservative justice Anthony Scalia in 2016, Mr McConnell refused to hold a vote to confirm a nominee put forward by then-President Barack Obama, a Democrat.