The American Civil Liberties Union is airing an advertisement in Kentucky this week to pressure Senate Majority Leader Mitch McConnell to hold a vote on criminal-justice reform backed by President Trump.

The 60-second ad says there is significant bipartisan support for the First Step Act, which would reduce federal mandatory minimum prison sentences for some drug offenses, but Mr. McConnell is dragging his feet.

“The common-sense agreement would give inmates who have served their time a second chance at life,” the ad says. “It has the support of stalwart conservatives like Chuck Grassley and Mike Lee — and progressives like Dick Durbin and Cory Booker. There’s one person stopping it: Senator Majority Leader Mitch McConnell.”

Mr. McConnell has said he will conduct a whip count to ensure the bill can pass, and he intends to hold discussions with colleagues on the measure this week. He has warned that the Senate has other priorities, including a Dec. 7 funding deadline to keep portions of the federal government open.

But the support among Republicans is not unanimous.

“The bill has got a lot of problems, and we need to hold a hearing,” said Sen. John Kennedy, Louisiana Republican.

“I don’t know if we are going to have time. I mean when you look at what we have on our plate, we got a bunch of nominations, we got a farm bill, we got a budget,” he added.

Sen. Tom Cotton, Arkansas Republican, is also not a fan of rushing through the bill.

He penned an op-ed for USA Today earlier this month, saying considering criminal justice reform during a lame-duck session is a “misguided effort to let serious felons out of prison.”

Mr. Trump, though, is urging Mr. McConnell to bring up the prison-reform measure for a vote in Congress’s last three working weeks of the year.

“We’re talking to him, and we’re doing a [vote] count,” the president said Monday night in Biloxi, Mississippi. “But from everything that we’re looking at right now, we have more than enough. So, at a certain point, we’ll have a talk.”

Asked if he has any leverage with Mr. McConnell, the president replied, “Well, I don’t need leverage. Look, he’s a friend of mine. He’s a really good guy. I think he believes in this, also.”

At a forum on the legislation with state and federal leaders in Mississippi, the president said the proposal “will help former inmates reenter society as productive law-abiding citizens, and it has tremendous support no matter where we go.”

“Communities will be safer by helping inmates gain the skills that they need to obtain jobs and stay out of prison after they are released,” he said.

Presidential adviser Jared Kushner, who has been the lead White House negotiator with Congress on the measure, said he’s “very confident that hopefully we’ll get this passed.”

“I think we all want the same thing,” Mr. Kushner said. “We want safer communities. We want people to have better opportunities and a better life.”

Sen. Roger Wicker, Mississippi Republican, said a “few tweaks might have to be made” to a House version of the legislation. But he said there will be “a real effort … to have this as an accomplishment of this Congress and this administration this year.”

But Sen. Lindsey Graham, South Carolina Republican, sounded slightly less optimistic.

“Jared, if this ever happens, it’d be because of your dogged determination,” he told Mr. Kushner at the forum. “And when you jumped onboard a couple weeks ago, Mr. President, everything fell in place. So I’m hoping, if Chuck [Senate Democratic leader Charles Schumer] and Mitch [McConnell] are listening, let’s vote in the next couple of weeks. Let’s end 2018 on a good note.”

The bill lost some GOP support after sentencing provisions were added to the First Step Act, which originally focused primarily on prison reform measures.

“The premise of this entire bill is our federal sentencing laws are unfair and that has not been demonstrated to me,” Mr. Kennedy told The Washington Times. “I want to have a debate about the sentencing laws.”

The First Step Act originally focused on corrections and had passed in the House overwhelmingly, with a 360-59 vote in May. But since then, a bipartisan group of senators added several sentencing provisions into the legislation.

The bill allows judges to sentence below mandatory minimums for certain nonviolent, drug offenders and rewards time credits to low-risk prisoners who participate in recidivism-reduction programs.

It also eliminates enhanced sentencing for an offender with a first-time offense who carries a firearm, instead applying enhanced mandatory sentences to offenders who had previously been convicted and did time for a violent crime or drug offense.

“Those are the real ones that have the teeth,” said Inimai Chettiar, the director of Brennan Center Justice Program.

“It’s very modest. The bill is not going to be opening the doors and cutting the prison population,” she added.

The legislation also prohibits prison officials from using restraints on pregnant prisoners, except those who are a flight risk, and it requires inmates to be housed no more than 500 miles away from their home.

• Alex Swoyer contributed to this report.

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