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.

The proposal would salvage some clean-energy tax credits and phase out others more slowly, making up some of the cost by imposing deeper cuts to Medicaid than the House-passed bill would.