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.

Senate Republicans on Monday proposed steeper Medicaid cuts than those included in the House-passed version of President Donald Trump’s signature policy bill—making the bill’s passage by the July 4 deadline set by Congressional Republicans even more uncertain as Medicaid cuts have been a major sticking point for some in the GOP.