“I think it is a sign that progress is being made, that we are getting there,” said Representative Steve Driehaus of Ohio, one of the Democrats who forced the House to adopt the stricter restrictions in its bill. The hospitals’ statement, he said, recognized the Senate’s compromise as a meaningful step, making him “optimistic” that Democrats could find a bill that he and other abortion foes could support.

Other abortion opponents argue that liberals are overstating the hospital association’s influence. “They don’t carry the same sway,” said Representative Bart Stupak, the Michigan Democrat who led the effort that resulted in the House bill’s including a full ban on abortion coverage in any subsidized health insurance plan.

Mr. Stupak said he still had commitments from at least 10 Democrats who voted for the House bill and pledged to vote against the final legislation if it loosened the abortion restrictions  enough to keep the bill from being approved. “At the end of the day we are going to have something along the lines of my language,” he said. Abortion rights supporters said the signs of openness from Catholic groups were helping some Democratic abortion foes accept the Senate compromise.

“We have known for quite some time that the Catholic hospitals and also the nuns are really breaking from these hard-line bishops and saying, ‘This really is our goal: to get more people into health care coverage,’ ” said Representative Diana DeGette, Democrat of Colorado.

The abortion rights faction of the House Democrats was initially dubious about the Senate bill’s provision but has warmed up to it after reassurances from their Senate counterparts, Ms. DeGette said. President Obama and Democratic Congressional leaders say they aim to follow 30-year-old rules blocking the use of federal money for elective abortions, but lawmakers have fiercely disagreed over how to do so.

Like most Catholic groups, the Catholic Hospital Association has echoed the bishops’ opposition to any federal financing of abortion in health care proposals. But its officials also stood at the White House last spring to endorse Mr. Obama’s plans as part of an administration deal with the hospital industry.

After the Catholic Hospital Association’s endorsement of the proposed compromise, Catholic conservatives and some abortion opponents accused the group of selling out to the Democrats.