"I will be putting forth a nominee next week. It will be a woman," Trump said during a rally in Fayetteville, North Carolina.
Trump told rallygoers that Ginsburg's "landmark rulings, fierce devotion to justice and her courageous battle against cancer inspire all Americans," offering prayers to the justice's family.
The President also noted her friendship with the late Supreme Court Justice Antonin Scalia, saying it's "a powerful reminder we can disagree on fundamental issues, while treating each other with decency, dignity and respect."
He also discussed how he has the power to fill her vacant seat.
"So, Article 2 of the Constitution says that the President shall nominate justices of the Supreme Court. I don't think it can be any more clear, can it?" Trump asked, prompting chants of "Fill that seat!" from the rally crowd.
The President told reporters earlier Saturday there about 45 people on his list, but he does have a "short list" for potential nominees.
"We want to respect the process, and the process will move, I think it is going to move very quickly actually. I agree with the statement put out by Mitch McConnell. I agree with it actually 100%. I put out a very similar statement you saw, so I think we're going to start the process extremely soon, and we'll have a nominee very soon," Trump told reporters before departing the White House on Saturday.
"I think the choice will be next week, I do," Trump later added.
Trump and McConnell spoke by phone Friday, a source familiar later confirmed to CNN on Saturday. Trump raised the names of Barrett and Barbara Lagoa as potential nominees during his call with McConnell, according to two people familiar with the conversation. Trump raised them unsolicited and McConnell didn't weigh in on either or talk about his personal preferences, the sources said.
Lagoa has the backing of many evangelicals while Trump lawyer Pat Cipillone has shown support for Barrett.
According to a source familiar with the internal conversations, naming an announcement is dependent on when Ginsburg is buried. Allies have raised concern about naming a replacement before she has had a proper burial, two sources told CNN.
White House communications director Alyssa Farah told Fox News on Saturday that the administration will move in "Trump time" on a Supreme Court nominee.
"Well, whether we have a vote or not is really a question for the Senate and for Leader (Mitch) McConnell, but it is fully this President's intention to do his job under the Constitution and to appoint someone to the seat," Farah said, also claiming that the date of the November 3 election is "irrelevant" to the President's desire to nominate someone.
When it comes to determining who will be the replacement -- one source said conversations are revolving around three main issues -- are they a committed conservative and textualist, do they have any Brett Kavanaugh issues, and will they cause problems for conservatives down the road by evolving like Chief Justice John Roberts.
Whatever happens, one of the sources involved said that everyone knows how ugly this process is going to get. "This is going to make the Kavanaugh hearings look like a pillow fight," the source said.
In a message to GOP senators on Friday, McConnell did not indicate a timeframe for considering the nomination but made clear he believes there's enough time to take up the nominee this year. While nominees typically take two to three months to process, McConnell, who controls the majority of the Senate, could fast track the process.
Conservatives are pushing McConnell behind the scenes to consider moving before the November election to fill the vacancy -- potentially leaving the conference divided over what timeline is best.
Republican Sen. Ted Cruz, who sits on the Senate Judiciary Committee, told Fox News on Friday that he believes Trump should nominate a successor to the court next week.
Ultimately, it will be McConnell's decision and not one he'll make without input from his members in the coming days.
In speaking with reporters Saturday morning, White House press secretary Kayleigh McEnany was asked by CNN's Joe Johns if the President has been in contact with McConnell or other Senate Republicans on next steps.
"Not that I'm aware of," she replied, adding, "Right now the President is squarely focused on honouring the legacy of Justice Ruth Bader Ginsburg."
McEnany also called on Biden to release a list of potential Supreme Court nominees so voters know where he stands.
Before news of Ginsburg's passing, Biden told reporters Friday that he doesn't think he will release any names of potential Supreme Court nominees before the election.
This story has been updated with additional developments Saturday.
CNN's Kristen Holmes, Ryan Nobles, Phil Mattingly, Sam Fossum, Joe Johns, Lauren Fox, Jason Hoffman, Kevin Liptak, Kaitlan Collins, Paul LeBlanc, Maegan Vazquez, DJ Judd and Nicky Robertson contributed to this report.