Donald Trump has pledged to nominate a woman to replace Ruth Bader Ginsburg on the US Supreme Court.
The president said he would choose his nominee next week and the process would move "quickly".
At a freewheeling rally in North Carolina on Saturday night supporters of the president chanted "Fill That Seat".
Mr Trump said: "I will be putting forth a nominee next week. It will be a woman."
He added: "I think it's going to move quickly actually. We want to respect the process, and I think it’s going to move very quickly. I think we’re going to start the process extremely soon and we’ll have a nominee very soon."
At the rally Mr Trump took an impromptu poll from the crowd, asking them to cheer for either a woman or a man to be his pick. The crowd cheered considerably louder for a woman.
He said: "That's a very accurate poll because that's the way I feel. It will be a woman. A very talented, very brilliant woman, who I haven't chosen yet, but we have numerous women on the list."
Mr Trump praised Justice Ginsburg, who died from cancer on Friday, as a "legal giant".
The president said:" Her landmark rulings, fierce devotion to justice and her courageous battle against cancer inspire all Americans."
Democrats have called for the decision to be delayed until after the election as it is so close, less than 50 days away. But Mr Trump said it was his constitutional right to install a successor and it would happen before the inauguration in January.
He said: "We have plenty of time. You’re talking about January 20."
Mr Trump praised two senior female judges as possible replacements - Amy Coney Barrett of Chicago, and Atlanta-based Barbara Lagoa.
His nominee would require approval by only a simple majority in the Senate, and Republicans hold a 53-47 majority. But not all Republican senators supported the decision to move quickly.
Susan Collins, the Republican senator from Maine, said it should wait until after the election.
She said: "In fairness to the American people, who will either be re-electing the president or selecting a new one, the decision on a lifetime appointment to the Supreme Court should be made by the president who is elected on November 3rd."
Senator Lisa Murkowski of Alaska, a Republican, said on Friday, just before Justice Ginsburg's death, that she would not vote for a Supreme Court nominee so close to the election.
Justice Ginsberg was the senior liberal on the nine-member court. Mr Trump's nominee would mean the court having a 6-3 conservative majority, which could have a major impact on issues including the legality of abortion.