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.

Senate Republicans are taking Medicaid cuts even further than their House counterparts with legislation that targets hospital payments.

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