Speaking during a campaign stop in Swanton, Ohio Trump promised enthusiastic supporters he would appoint a woman to fill the open seat.
Two federal appeals court judges appointed by Trump are clear front-runners: Amy Coney Barrett of the Chicago-based 7th U.S. Circuit Court of Appeals and Barbara Lagoa of the Atlanta-based 11th U.S. Circuit Court of Appeals. Trump met with Barrett at the White House on Monday, according to a source familiar with the situation.
Ginsburg died on Friday of complications from pancreatic cancer at age 87.
Trump's announcement would come before Ginsburg is due to be buried privately at Arlington National Cemetery next week.
Officials have arranged for a public viewing of her body outside the Supreme Court on Wednesday and Thursday and inside the U.S. Capitol on Friday.