WASHINGTON — Senate leaders agreed yesterday that they would wait until June to take final action on a bipartisan plan to give millions of unlawful immigrants legal status.

The measure, which also tightens border security and workplace enforcement measures, unites a group of influential liberals, centrists, and conservatives and has White House backing, but it has drawn criticism from across the political spectrum. In a nod to that opposition, Senate leaders won’t seek to complete it before a hoped-for Memorial Day deadline.

“It would be to the best interests of the Senate … that we not try to finish this bill this week,” said Senate Majority Leader Harry Reid, a Democrat from Nevada, as the chamber began debate on the volatile issue. “I think we could, but I’m afraid the conclusion wouldn’t be anything that anyone wanted.”

The bipartisan compromise cleared its first hurdle yesterday with a bipartisan Senate vote to begin debate on a separate immigration measure. Still, it faces significant obstacles as lawmakers seek dozens of modifications to its key elements.

Republicans want to make the bill tougher on the nation’s estimated 12 million illegal immigrants. Democrats want to change a new temporary worker program and reorder priorities in a merit-based system for future immigration that weights employability over family ties.

The unlikely coalition that brokered the deal, led by Senators Kennedy, a Democrat from Massachusetts, and Kyl, a Republican from Arizona, is plotting to protect the agreement from “deal-breaker” changes that would sap its support. The group will hold daily meetings starting Tuesday to determine whether proposed revisions would sink what they are calling their “grand bargain.”

“We have to try our very best to work together to get something that will actually pass,” Mr. Kyl said.