WASHINGTON—Senate GOP leaders picked up support Wednesday for their plan to pass a scaled-back bill to repeal a handful of elements in the current health law, and then open negotiations with House Republicans to try to bring together their two very different bills.

Republican senators said they recognized passing a “skinny repeal” would essentially postpone tough decisions on health care until later, but they seized on it as potentially their best option as the Senate this week began considering and rapidly discarding other...