WASHINGTON—Congressional leaders reached a bipartisan deal late Sunday to fund the government until October, while Republicans scrambled to pull off an even more significant legislative achievement that has eluded them this year: an overhaul of the health-care system.

Both President Donald Trump and Vice President Mike Pence in television interviews Sunday suggested confidence that they could win enough votes to pass a bill to undo the Affordable Care Act. But skepticism among centrist members of the party remains a stumbling...