A lawmaker who helped negotiate a bipartisan bill to overhaul immigration predicted on Sunday that comprehensive legislation would overwhelmingly pass the Senate by July 4 while House Republicans cautioned that they would write their own version, one piece at a time.

Sen. Chuck Schumer, D-N.Y., said he anticipates as many as 70 of the 100 senators will vote for the measure heading to the full Senate on June 10. Even if it passes there, the proposal faces tough prospects in the Republican-led House, where lawmakers are at work on their own piecemeal approach that could stall a pathway to citizenship for the estimated 11 million immigrants living in this country illegally.