WASHINGTON—Several of Republicans’ largest proposed spending cuts can’t be done in the fast-track budget process they are using to advance their megabill, the Senate parliamentarian ruled, dealing a significant blow to the GOP’s hopes of passing their plans quickly.

A new proposal to help stabilize rural hospital access is floating around Capitol Hill as Senate Republicans consider their reconciliation bill.

Republicans are struggling to devise a solution to the health care problem their package has created. Already, estimates say 10.9 million more people would be without health coverage under the House-passed version of the bill. GOP senators have proposed steeper reductions, which some say go too far.