The moves signal the start of a monumental Senate confirmation fight over objections from Democrats that the appointment would come too close to the November election.
Mr Trump said he expected to announce his pick by the end of the week, before the court’s liberal icon Ms Ginsburg is buried at Arlington National Cemetery.
The president told reporters he was still going to be interviewing other candidates and might meet with Judge Barbara Lagoa when he travels to Florida later this week.
But Ms Barrett has long been favoured by conservatives, and those familiar with the process said interest inside the White House seemed to be waning for Ms Lagoa amid concerns by some that she did not have a proven record as a conservative jurist.
Democrats, led by presidential nominee Joe Biden, are protesting the Republicans’ rush to replace Ms Ginsburg, saying voters should speak first, on election day on November 3, and the winner of the White House should fill the vacancy.
Mr Trump dismissed those arguments, telling Fox & Friends: “I think that would be good for the Republican Party, and I think it would be good for everybody to get it over with.”
The mounting clash over the vacant seat — when to fill it and with whom — injects new turbulence in the presidential campaign with the nation still reeling from the coronavirus pandemic that has killed nearly 200,000 Americans, left millions unemployed and heightened partisan tensions and anger.
Up until now, the race has been largely a referendum on how Mr Trump has managed or mismanaged the Covid-19 pandemic.
Democrats point to hypocrisy in Republicans trying to rush through a pick so close to the election after Senate Majority Leader Mitch McConnell led the GOP in refusing to vote on a nominee of President Barack Obama in February 2016, long before that year’s election.
Mr Biden is appealing to GOP senators to “uphold your constitutional duty, your conscience” and wait until after the election.
Conversations in the White House and Mr McConnell’s office have been increasingly focused on Ms Barrett and Ms Lagoa, according to a person granted anonymity to discuss the private deliberations.
An appellate court judge, Ms Barrett was a strong contender for the seat that eventually went to Brett Kavanaugh in 2018. At the time, Mr Trump told confidants he was “saving” Ms Barrett for Ms Ginsburg’s seat.
Ms Lagoa has been pushed by some aides who tout her political advantages of being Hispanic and hailing from the key political battleground state of Florida.
Mr Trump said he was planning to name his pick by Friday or Saturday, ahead of the first presidential election debate. With little more than a month before the election, Mr McConnell said the Senate had “more than sufficient time”.
No nominee has won confirmation so quickly since Sandra Day O’Connor — with no opposition from either party — became the first woman to serve on the Supreme Court in 1981.
Ms Ginsburg, 87, died on Friday of metastatic pancreatic cancer. She will lie in state at the US Capitol this week, the first woman ever accorded that honour. First, her casket is to be on view mid-week on the steps of the high court.
She is to be buried next week in a private service at Arlington National Cemetery.
