The sources describe the list of potential nominees as very short and including at least one woman. U.S. Circuit Judge Amy Coney Barrett is seen as a leading contender, two sources tell ABC News.
Trump was asked about Ginsburg's death after leaving his Friday night campaign rally in Minnesota and said he was not aware of her passing. The 87-year-old justice died Friday after a battle with pancreatic cancer, the court announced.
"Wow. I didn’t know that. I just -- you’re telling me now for the first time," he told reporters. "She led an amazing life. What else can you say? She was an amazing woman. Whether you agree or not, she was an amazing woman who led an amazing life. I’m actually sad to hear that. I am sad to hear that."
Sources added that Senate Majority Leader Mitch McConnell has already been in touch with members of the Republican caucus after news of Ginsburg’s passing was announced. The Senate needs just a simple majority to confirm a nominee, with Republicans currently holding 53 seats.
McConnell refused to bring then-President Barack Obama's nominee to replace Antonin Scalia in 2016 -- Merrick Garland -- to the floor for a vote, but said he will not do the same this time.
"In the last midterm election before Justice Scalia’s death in 2016, Americans elected a Republican Senate majority because we pledged to check and balance the last days of a lame-duck president’s second term. We kept our promise. Since the 1880s, no Senate has confirmed an opposite-party president’s Supreme Court nominee in a presidential election year," McConnell said in a statement following Ginsburg's death.
"By contrast, Americans reelected our majority in 2016 and expanded it in 2018 because we pledged to work with President Trump and support his agenda, particularly his outstanding appointments to the federal judiciary. Once again, we will keep our promise," he continued. "President Trump’s nominee will receive a vote on the floor of the United States Senate."
Trump told CNN in March 2016 that he believed the next president -- presumably him -- should pick the nominee, not Obama.
"I think the next president should make the pick, and I think they shouldn't go forward," he told "New Day" on March 16, 2016. "And I believe I'm pretty much in line with what the Republicans are saying. I think that the next president should make the pick. We don't have a very long distance to wait."
Sen. Lindsey Graham, R-S.C., went as far as to say Republicans had officially changed procedures in a speech from the floor in 2016 and to play back the video if a Supreme Court spot came up in 2020.
"If there’s a Republican president in 2016 and a vacancy occurs in the last year of the first term, you can say Lindsey Graham said let’s let the next president, whoever it might be, make that nomination, and you can use my words against me and you’d be absolutely right," he said. "We are setting a precedent here today."
Obama released a statement both extolling the virtues of Ginsburg, while simultaneously reminding Republicans of their decision in 2016.
"Four and a half years ago, when Republicans refused to hold a hearing or an up-or-down vote on Merrick Garland, they invented the principle that the Senate shouldn’t fill an open seat on the Supreme Court before a new president was sworn in," he said in a statement. "A basic principle of the law -- and of everyday fairness -- is that we apply rules with consistency, and not based on what’s convenient or advantageous in the moment."
"The rule of law, the legitimacy of our courts, the fundamental workings of our democracy all depend on that basic principle," he continued. "As votes are already being cast in this election, Republican Senators are now called to apply that standard.
Current Democratic presidential nominee Joe Biden was vice president at the time of Garland's nomination and told reporters in Delaware late Friday on his return from campaigning in Minnesota that Republicans should stick to what they said in 2016.
"In the coming days, we should focus on the loss of justice, and her enduring legacy. But there is no doubt -- let me be clear that the voters should pick the president, and the president should pick the justice for the Senate to consider," Biden said. "This was the position the Republican Senate took in 2016 when there were almost 10 months to go before the election. That's the position the United States Senate must take today."
Even if a nominee is put forth, the timeframe would seem tight for confirming a justice before Election Day. The average number of days from SCOTUS nomination to final vote in the Senate is 69.6 days -- about 2.3 months -- according to the Congressional Research Service.
There is ample precedent for nominations and confirmations to the Supreme Court in presidential election years. It's happened six times since 1900. The most recent nomination and confirmation in an election year was 1940, after Justice Pierce Butler died in office and Franklin D. Roosevelt nominated Frank Murphy in January 1940; he was confirmed 12 days later.
The latest election year confirmation came in 1916 when Charles Evans Hughes resigned in June and President Woodrow Wilson nominated John Clarke on July 14. Ten days later he was unanimously confirmed. There has never been one filled later than that ahead of an election.
Given all of this, the 2016 GOP blockade of Garland was truly an anomaly, one which McConnell now seems keen to keep that way if given the chance.
Swing Sens. Lisa Murkowski, R-Alaska, and Susan Collins, R-Maine, have each recently told media they do not believe a vote should be taken. Murkowski told Alaska Public Radio on Friday, prior to Ginsburg's death, that no votes should be taken, while Collins told The New York Times' Jonathon Martin earlier this month that it was "too close" to seat a justice in October.
Trump allies have already lined up behind the idea of quickly confirming a replacement. Texas Sen. Ted Cruz told Fox News' Sean Hannity, "I believe the president should next week nominate a successor to the court. I think it's critical the Senate takes up and confirms the successor before Election Day."
Barrett, a former Notre Dame law professor who previously clerked for Scalia, was confirmed to the 7th U.S. Circuit Court of Appeals in Chicago in October 2017.
Here's a list of other potential nominees based on ABC News reporting:
Judge Joan Larsen
Larsen was confirmed to the 6th Circuit in Cincinnati in October 2017 and previously served on the Michigan Supreme Court. She is a University of Michigan law professor, and, like Barrett, she also clerked for Scalia.
Judge Amul Thapar
Thapar was confirmed to the 6th Circuit in May 2017 and was previously federal judge in Kentucky before Trump nominated him to the 6th Circuit.
Judge Raymond Kethledge
Kethledge was confirmed to the Court of Appeals for the 6th Circuit in 2008 after being nominated by President George W. Bush. He is a former clerk for Justice Anthony M. Kennedy and a former in-house lawyer at Ford Motor.
Judge Britt Grant
Grant was confirmed last July to the 11th Circuit at Atlanta. She previously was a Georgia Supreme Court justice, a Georgia solicitor general and a partner at Kirkland & Ellis.
Judge Thomas Hardiman
Hardiman, of the 3rd Circuit at Philadelphia, was the first person in his family to attend college, and he helped pay for his Georgetown University law degree by driving a taxi.
He was also a top contender for the first two vacancies under President Trump.
Judge Neomi Rao
Rao, a former clerk to Justice Clarence Thomas, is a Trump appointee who holds the seat previously occupied by Brett M. Kavanaugh on the U.S. Court of Appeals for the D.C. Circuit. She recently wrote the opinion ordering the dismissal of the case prosecuting Michael Flynn.
ABC News' Karma Allen, Mark Osborne, Devin Dwyer, Molly Nagle, Will Steakin, Trish Turner and Chris Donovan contributed to this report.