WASHINGTON—Senate Republicans’ plan to extract more savings from Medicaid as part of their revised tax and spending package triggered a backlash within the party, raising concerns that leaders could miss their self-imposed deadline of getting the bill to President Trump’s desk by July 4.

Rep. Mike Lawler, a New York Republican, issued a stern warning to Senate Republicans earlier on Monday: Any changes to pare back the deal, he said, would cause the bill to collapse in the House.