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...