WASHINGTON—Senate Majority Leader Mitch McConnell (R., Ky.) blocked a bill designed to protect special counsel Robert Mueller from being fired, but the effort’s bipartisan sponsors said they would keep pressing the matter.

The resignation of Jeff Sessions as attorney general at the president’s request last week and the appointment of Acting Attorney General Matthew Whitaker increased calls from Republicans and Democratic senators to pass legislation to protect the special counsel’s continuing investigation into Russian interference...