With the exception of two Republican members of the bipartisan group who also sit on the Judiciary Committee — Senators Jeff Flake of Arizona and Lindsey Graham of South Carolina — Mr. Hatch is considered the other Republican member most likely to support the bill, which would give it some conservative gravitas as it heads to the Senate floor.

The bipartisan group is hoping for a large Senate vote in support of the bill — some members have suggested as many as 70 votes — to place pressure on House Republicans to also get behind the legislation. And Mr. Hatch’s “yes” vote is seen as a driver for what they hope will be a groundswell of support. Mr. Hatch, who hails from a state with a dynamic immigration culture, meanwhile finds himself with a final chance to help pass a broad immigration overhaul, something in which he has both publicly and privately expressed a genuine interest.

But Mr. Hatch’s “yes” vote in committee comes with a major caveat — the acceptance of provisions of his that would, among other things, help technology companies by increasing the number of temporary visas available for high-skilled workers (known as H-1B visas) and ease restrictions around the hiring and firing of workers. He also is offering some provisions on the finance side, including one that would require immigrants to show they have paid back taxes and are staying current with them as they proceed toward legal status.

“I personally believe we need immigration reform,” said Mr. Hatch, adding that his tweaks to the bill are “amendments that clean it up and make it a better bill, and may even be able to get it more votes.” The provisions are also, he explained, an all-but-ironclad requirement for earning his vote: “They know my point is I’ve got to have those,” he said, referring to the message he has communicated to the group.

And so last week, as the committee met twice to discuss the bill publicly, the behind-the-scenes negotiating and arm-twisting picked up in earnest, with Mr. Schumer’s office taking the lead in trying to work out an agreement with Mr. Hatch.