WASHINGTON—Senate Republicans unveiled a revised health bill Thursday in an effort to shore up its faltering GOP support, but its future remained precarious as at least two Republicans quickly vowed to oppose it, bringing the proposal within a single vote of potential collapse.

Senate Majority Leader Mitch McConnell (R., Ky.) implored his fellow Republican lawmakers to at least support a procedural measure next week that would allow debate to begin on the bill, which aims to topple large parts of the Affordable Care Act.

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