“I don’t believe we should allow thousands of violent felons to be released early from prison, nor do I believe we should reduce sentences for violent offenders in the future,” Mr. Cotton said in an interview. He said that the criminal justice legislation was being driven by a “myth” of mass incarceration of low-level, nonviolent offenders in federal prisons, and that most prisoners had already cut their sentences under plea bargains.

Some senators said Mr. Cornyn and other Republican backers of the legislation may have misread the willingness of their colleagues to embrace the criminal justice movement and had too much faith that others would be swayed by the support of conservatives such as Senators Charles E. Grassley of Iowa, the chairman of the Judiciary Committee, and Mike Lee of Utah.

“John has some work to do, big-time work,” to secure enough support to persuade Mr. McConnell to go forward, said one Republican senator who spoke on the condition of anonymity to discuss internal party matters.

Mr. Cornyn, who became enthusiastic about the issue after successful changes to the criminal justice system in Texas, where he was a judge for 13 years, said discussions had begun on the staff and senator-to-senator levels to clear up misconceptions. He noted that sentencing provisions in the measure would apply to about 5,000 offenders — far fewer than in earlier sentencing proposals — and that those now serving time would have to reappear before the same judge and prosecutor to win a reduction in time served.

“Some people are now realizing for the first time that they may have to vote on this thing, so they need more information,” said Mr. Cornyn, who added that he would be asking colleagues what changes might be needed to win their backing.

He said that Mr. McConnell would like to lessen the party divisions before deciding to devote weeks of scarce Senate floor time to the criminal justice proposition.

“This is all part of his calculation, and we are trying to work with him and all my colleagues and figure a way to get this done,” Mr. Cornyn said. “It is not the kiss of death if you don’t do this in 2016, but I would like to. I think we have a window here.”