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.

U.S. Senate Republicans' proposed changes to President Donald Trump's sweeping tax-cut and spending bill are expected to push U.S. debt levels higher, roughly in line with projections from the House-p...