WASHINGTON—Legislation to speed up the process for employment-based green cards for some Indian immigrants caught in a yearslong backlog could receive a vote in the Senate as early as this week, after the bill’s sponsor and a key objector reached a bipartisan compromise.

The bill would lift per-country caps on permanent residency for immigrants who have job offers in the U.S. That would allow people who have waited in the backlog the longest—primarily Indian and some Chinese immigrants on H-1B high-skill visas—to receive green cards first under a new system.

A version of the bill passed with a wide bipartisan majority in the House in July, and Sen. Mike Lee (R., Utah) has brought up his version in the Senate multiple times through a process known as unanimous consent, which allows bills to bypass roll-call votes. But Mr. Lee’s bill has drawn objections from one or more senators each time it was introduced.

Mr. Lee reached a compromise with one of the bill’s longest holdouts, Sen. Dick Durbin (D., Ill.), Senate aides said Tuesday night, reviving its chances of receiving a vote.

Under current U.S. law, no more than 7% of employment-based green cards can be issued to any one country, though in practice countries including India receive more because the leftover visas are reallocated to meet demand. Critics of that approach in both parties say allotting the same number of green cards to each country gives an advantage to those applying from smaller nations while penalizing Indian and Chinese applicants who qualify but must wait years for their turn because they are in larger pools of applicants.