Mr Trump indicated he would hold off on announcing his choice until the end of the week out of respect for the memorial services for Ginsburg taking place in the capital.
He has pledged to choose a woman for the lifetime post.
Leaders of the Republican majority in the Senate, which is tasked with confirming court nominees, said they have enough support to hold a vote on the nomination either before the election or at worst during the “lame-duck” session between the election and the inauguration of the next president in January.
“We will certainly do that this year,” Republican Senate leader Mitch McConnell said.
Although two Republican senators said they believed the upper chamber of Congress should not vote at all before the election, the party’s 53-47 majority is still just big enough to go ahead.
One of the other key potential Republican holdouts, frequent Trump critic Mitt Romney, said Tuesday he would move ahead with the process.
“If the nominee reaches the Senate floor, I intend to vote based upon their qualifications,” Mr Romney said.
Mr Trump welcomed what he called a “very positive statement” by Romney, who was the only Republican senator to vote for impeaching the president.
“Thank you Mitt,” he said.