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 proposals from Republicans keep in place the current $10,000 deduction of state and local taxes, called SALT, drawing quick blowback from GOP lawmakers from New York and other high-tax states.