Ginsburg, a liberal judge in the US Supreme Court, died at his home on Friday.
It was not long before discussions about when, and how, she would be replaced flared up.
According to CNN sources, Trump will nominate a replacement sometime before the first presidential debate on September 29.
Ginsburg's replacement may have a major influence over HD cases involving abortion and the right to vote, as it may determine the political majority in court.
If a Conservative judge takes Ginsburg's place, the court will have a solid Conservative majority, and it could be for decades.
"Counting on us"
New judges are nominated by the president and must be approved by a majority of Senate members.
But the fact that President Trump wants to nominate a judge the same year as the election gets harsh criticism from Democrats.
The party's presidential candidate Joe Biden wants a replacement to be proposed only after the election.
"There is no doubt that voters should be allowed to elect a president, and the president should elect a judge for the Senate to consider," he said in a statement.
Vice Presidential candidate Kamala Harris has also commented on the nomination:
"Millions of Americans are counting on us to win and defend the Supreme Court," she wrote on Twitter.
Thinking of letting the Senate vote
It has happened before that they have had to postpone nominating an HD judge.
When former President Barack Obama wanted to nominate a new judge in 2016, it was dismissed because it was an election year.
This time, however, Senate Majority Leader Republican Mitch McConnell has said he intends to let the Senate vote if Trump nominates a candidate.
But even if the Senate were to vote, it is not certain that Trump will get through his election.
The party has a small majority with 53 out of 100 seats, and it is not certain that the presidential nomination will appeal to all Republicans.