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.

President Trump’s political base is splintering over the prospect that the U.S. could join Israel’s assault on Iran—and the White House is trying to quell the backlash.