Trump said in an interview with Fox News he's narrowed his list of potential justices to five, and said a confirmation vote in the Senate should be taken before the election on Nov. 3.
Trump said he expects to announce his nominee on Friday or Saturday. He's previously said the nominee will be a woman.
Since Ginsburg's death on Friday, Democrats have called on Republicans to abide by the standard they set in 2016 after the death of Justice Antonin Scalia nine months before the election. Republicans then said the voters should choose a president, and that president should nominate a replacement for Scalia.
"The bottom line is we won the election, we have an obligation to do what's right and act as quickly as possible," Trump said Monday. "These are the smartest people, the smartest young people, you like to go young because they're there for a long time."
Federal appeals court judges Amy Coney Barrett and Barbara Logoa are said to be on Trump's list.
Coney Barrett serves on the Court of Appeals for the Seventh Circuit and Logoa on the Court of Appeals for the 11th Circuit. Another potential nominee, judge Allison Jones Rushing, sits on the Court of Appeals for the 4th Circuit.
Ginsburg, 87, died at her home after a battle with pancreatic cancer. She transcribed a request before her death that her successor not be decided until after the election.
At least two Republican senators -- Susan Collins of Maine and Lisa Murkowski of Alaska -- have already said they won't support replacing Ginsburg before the election. Democrats would need two more, for a total of 51 "no" votes in the chamber, to block Senate confirmation.