The divisions were on display on Thursday. At the Senate Republicans’ weekly caucus luncheon at the Capitol, Mr. McConnell acknowledged that the changes had influential supporters who had worked hard on the issue, but also invited two of its chief critics, Senators Tom Cotton of Arkansas and John Kennedy of Louisiana, to deliver remarks, two Republican congressional officials said.

Mr. Cotton, who has been perhaps the loudest critic of the bill’s sentencing changes in the Senate, urged colleagues to slow down the process, saying that the bill’s impact and implications were too expansive to push through without hearings, according to another official familiar with his remarks. He stressed opposition by some law enforcement groups and warned that a draft version of the bill he had seen would lead to the immediate release of thousands of felons onto the streets.

Mr. Cotton’s office also circulated a letter from groups representing elected sheriffs raising objections to the bill. Without changes, it said, the “legislation creates a high-risk path for dangerous criminals with gun crime histories to early release from prison.” Mr. Cotton also wrote an op-ed article in USA Today in which he called the legislation “a misguided effort to let serious felons out of prison.”

The legislation’s advocates caught a break last week when Mr. Trump fired his attorney general, Jeff Sessions, who had been one of the bill’s most vocal opponents within the administration.

Mr. Sessions’s temporary replacement, Matthew G. Whitaker, appears unlikely to exercise similar influence on the issue. In a recent phone call, Mr. Whitaker told Mr. Grassley that he would support the legislation backed by the president, according to George Hartmann, a spokesman for Mr. Grassley.

The legislation, the First Step Act, builds on a prison overhaul bill passed overwhelmingly by the House this year, adding four additional changes to federal sentencing laws. It combines new funding for anti-recidivism programs meant to better prepare inmates to re-enter society and the expansion of early-release credits for prisoners, and reduces some mandatory minimum sentences for nonviolent drug offenders, among other changes.

Though its effect would be limited to federal prisons and offenses — not state ones — experts believe the legislation could shape the experiences of tens of thousands of current inmates and future offenders.