The push comes despite ample apprehension from influential Republicans that the GOP could pay a political price for treating a nominee under Trump differently than they did under Obama. It also comes as Democrats are increasingly worried about the fragile health of Justice Ruth Bader Ginsburg, the 87-year-old liberal jurist who recently made public a new bout with cancer, and the possibility of other retirements.
"We will," said Sen. John Thune of South Dakota, the second-ranking Republican leader, when asked if the Senate would fill a vacancy, even during the lame-duck session after the presidential election. "That would be part of this year. We would move on it."
But the veteran Iowa Republican who chaired the Judiciary Committee in 2016 and helped block Judge Merrick Garland -- Obama's nominee -- by refusing to schedule election-year confirmation hearings, said he would not fill a fill a vacancy now for the same reason.
"My position is if I were chairman of the committee I couldn't move forward with it," Sen. Chuck Grassley told CNN.
The current Judiciary Committee chair, Trump ally Sen. Lindsey Graham of South Carolina, has professed differing views about whether he would try to confirm a nominee during the last year of Trump's term.
Asked about his past opposition to moving a nominee in a presidential election year after the primary season, Graham said: "After Kavanaugh, I have a different view of judges," referencing the brutal 2018 confirmation process of Supreme Court Justice Brett Kavanaugh.
"I'd like to fill a vacancy. But we'd have to see. I don't know how practical that would be," Graham told CNN Monday. "Let's see what the market would bear."
Sen. Josh Hawley, a Missouri Republican who's a member of the Judiciary Committee, said that if a vacancy were to occur, he would like to get a nominee confirmed before the court's term begins in October.
Hawley said he would be "shocked" if Trump didn't try to fill a vacancy despite GOP arguments in 2016 that voters should decide which president selects a nominee during an election year.
Hawley said the difference between then and now is that Obama couldn't run again but Trump is on the ballot trying to win a second term.
"I think we have a different set of circumstances. We have a President who is very actively running for reelection," Hawley said. "He's going to be on the ballot. People are going to be able to render a verdict on him like they couldn't on Obama. My guess is he would absolutely nominate somebody. I would be shocked if he didn't."
Republican Sen. Joni Ernst of Iowa, who is running for reelection, told Iowa PBS last week she supports confirming a potential nominee this year, according to The Des Moines Register.
"(If) it is a lame-duck session, I would support going ahead with any hearings that we might have," she said. "And if it comes to an appointment prior to the end of the year, I would be supportive of that."
Senate Majority Leader Mitch McConnell, a Kentucky Republican, has repeatedly vowed to fill a vacancy this year and has said the difference between now and 2016 is that by the time Obama, a Democrat, nominated Garland to fill the vacancy left by the death of Justice Antonin Scalia, Republicans controlled the Senate. Right now, Republicans control both the White House and the Senate.
A vacancy could put some GOP senators in a tough spot. Asked if he supported filling a vacancy this year, Sen. Thom Tillis, a vulnerable Republican running for reelection in North Carolina, said, "I am praying for Justice Ginsburg's health. That's all I'm really focused on right now."
Asked about filling a vacancy caused by retirement, not death, he downplayed the likelihood that would happen.
"I don't think there are many indications that there are. Normally those moves are made back in June over the session. I don't see any real possibility that there will be one," Tillis said.