WASHINGTON — Influential Republican senators, working in concert with the White House, are making a last-ditch effort to pressure Senator Mitch McConnell and other party leaders to take up a bipartisan criminal justice bill at risk of being shunted aside in the rush to complete other business before Congress adjourns.

With roughly two weeks remaining, senators and outside groups arguing for sentencing and prison law changes included in the bill, known as the First Step Act, have all but accused Republican leaders of underrepresenting the bill’s support to justify not allowing a vote. Supporters have additionally raised the possibility of attaching the long-sought changes to a must-pass year-end spending bill if leaders do not relent.

“There are a lot of people who don’t want this bill to come up, and they are using the whip organization to accomplish their goals,” said Senator Charles E. Grassley of Iowa, the Republican leader of the Judiciary Committee and the bill’s lead author, referring to the formal apparatus responsible for measuring support in the Senate. “But I know what I know.”

One of Mr. Grassley’s allies, Senator Mike Lee, Republican of Utah, said on Thursday that he had secured firm commitments of support from at least 28 Republicans, more than half of the caucus, and from all 49 Senate Democrats — well over the 60 to 70 votes party leaders indicated would be necessary for consideration. But Republican leaders insist that their own vote counting shows substantially less support, and that as long as they do not have a majority of Republicans on board, they will not advance a bill with far-reaching implications for the nation’s criminal justice system.