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 on Monday released its version of the GOP’s package that calls for enacting sweeping cuts to Medicaid and preventing a multi-trillion dollar tax hike on Americans.