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.

Republicans are seeking to limit a tax loophole that gives states more federal matching funds. Many state budgets could be hurt.

The Senate Finance Committee is revising the proposed Medicaid cuts in the GOP’s “big beautiful bill” — a move sure to spur vehement opposition from hospitals.