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.

“This is going to be jet fuel,” House Speaker Mike Johnson said on NBC’s “Meet the Press” earlier this month.

Senate Majority Leader John Thune, who can afford to lose only three GOP votes on Trump’s final tax and spending bill.