Vice President Mike Pence had a guest with him at Tuesday’s Senate Republican lunch: White House adviser and presidential son-in-law Jared Kushner. It was a sign the Trump administration is leaning on GOP leaders to schedule floor time for a key priority — overhauling the criminal justice system.

Kushner has been helping to lead the administration’s advocacy for a criminal justice overhaul bill that supporters are trying to cajole Senate Majority Leader Mitch McConnell to put on the floor before the end of the lame-duck session of Congress.

According to supporters of the bipartisan measure, it was Pence who made much of the sales pitch Tuesday.

“The vice president expressed his and the president’s support for the bill, and we had a long discussion over the bill,” said McConnell’s fellow Kentucky Republican, Sen. Rand Paul. “I think the majority of people who spoke on the bill are in favor of passing the bill.”

Sen. Tom Cotton of Arkansas, a vocal critic of the measure, expressed his fears that sex offenders could qualify for credits toward early release.

But GOP members of the bipartisan coalition behind the legislation were upbeat after Tuesday’s lunch.

“I’m feeling good,” South Carolina Sen. Lindsey Graham said. “We’ve got a few tweaks we could make, and the whole point is to deal with the recidivism problem.”

“The ‘good time’ credits are designed to keep people calm in jail so they don’t blow the place up. There are no programs focused that have credits behind them so that you have a skill set to stay out of jail,” added Graham, likely the next Judiciary chairman. “What we’re doing is creating a new concept, an earned credit that would be focused on skill sets.”

Graham and Paul are not always on the same side, but they are when it comes to the criminal justice and sentencing legislation. Senate Majority Whip John Cornyn of Texas has said an upcoming GOP whip count would gauge the current level of support.

“I’ve heard that the whip count’s at least half right now are favoring, or more, of Republicans,” Paul said mid-afternoon Tuesday.

Paul has suggested 60 to 70 senators would vote for the bill without further tweaks, and he told Roll Call on Tuesday that he didn’t give much credence to concerns about needing a robust amendment process.

“There’s all kinds of gamesmanship that they usually have, to avoid amendments, so no, I don’t think anything will be added that will torpedo the bill,” he said.

Paul’s often lonely efforts to offer amendments to legislation have regularly been thwarted by leadership, and it is ultimately McConnell’s decision what reaches the floor.

Cotton has been warning of grave consequences if the bill becomes law, something the authors have sought to refute.

“It’s almost a guaranteed unfortunate reality that when you release thousands of violent serious and repeat offenders within weeks or months of the bill passing, that some of them are going to commit violent crimes once they’re released that they would not have committed if they were still serving the sentence to which they were sentenced years ago,” the Arkansas Republican told talk radio host Hugh Hewitt.

Asked about Cotton’s opposition, Graham said Tuesday, “If you’ve ever looked at three strikes and you’re out and you’re OK with the way it’s worked, then you’re just looking at a different movie than I am.”

Support across the aisle

The bill has broad support among Senate Democrats, and it was discussed during Tuesday’s Senate Democratic caucus meeting. Connecticut Sen. Christopher S. Murphy said the question now is whether McConnell wants to take the time to work around the opposition of Cotton and like-minded senators.

“If you want to get criminal justice reform, you’ve got to find a way to get around Tom Cotton,” Murphy said. “That’s been the case from Day One.”

It is not a given that the measure makes it to the floor.

McConnell wants to continue confirming judicial and other nominations, with Thomas Farr’s nomination to be a district judge in North Carolina up for a procedural vote as early as Wednesday.

Senate Minority Leader Charles E. Schumer of New York and the entire Democratic Conference, as well as civil rights groups, want to keep Farr from the bench.

Among other criticisms, they point to how the Raleigh-based lawyer, who is white, defended in court North Carolina’s voting laws that judges later struck down as discriminatory for targeting minorities “with almost surgical precision.”

Farr’s fate is unclear. Part of the reason for that is that retiring Arizona GOP Sen. Jeff Flake has said he will oppose judicial nominations until he secures a vote on legislation to shield special counsel Robert S. Mueller III from an improper dismissal before his Russian election interference probe is finished.

Flake is scheduled to join Democratic Sens. Cory Booker of New Jersey and Chris Coons of Delaware to make another unanimous consent request to get the bill called up Wednesday.

But aside from the floor theatrics, Cornyn suggested Tuesday that negotiations with Flake could take place in the interest of getting more  judicial nominees cleared in the closing weeks of this Congress.

“We’re going to be working with Sen. Flake to see what he needs in order to lift his hold,” Cornyn said on Hewitt’s radio program. “There is a possibility we will have a vote on the Mueller, so-called Mueller protection bill, but I think there really is some serious constitutional issues on that, and I certainly don’t support it.”

Todd Ruger contributed to this report.