WASHINGTON — Democratic leaders persuaded four more House members Friday to support a landmark health-care bill after initially opposing it. But they still need votes from a pivotal bloc of lawmakers who remain concerned that the proposal would open the door for the federal funding of abortions.

More than 200 House members have announced they will vote Sunday against the Senate’s health-care bill. That leaves Democrats little margin for error as they try to gather the 216 votes needed for passage among the few dozen lawmakers who remain publicly undeclared.

Speaking to thousands of cheering supporters Friday at George Mason University in Fairfax, Va., President Obama predicted a “tough vote” as he prepared to meet with House Democrats in a final rally at the Capitol today. House leaders publicly predicted victory, but they kept private their own vote count as they continued to woo the undeclared Democrats.

The biggest target of that lobbying effort are anti-abortion Democrats; garnering their votes would ensure the bill’s passage, senior Democrats said Friday.

Those holdout lawmakers, most of whom hail from the Midwest and are Catholic, generally support the $940 billion package and its aim of providing coverage for 32 million more Americans. But they have voiced objections to how the Senate bill would handle insurance coverage of abortions.

“I want to vote for the bill. I just need to take care of a few issues before I can,” Rep. James Oberstar, D-Minn., said Friday.

In recent days, abortion opponents have gathered in meeting rooms and in huddles on the House floor, trading ideas and proposals, meeting with clergy and other religious officials who are also deeply divided over the impact of the Senate bill’s language.

One potential compromise could include staging a vote, separate from the health bill, on stronger anti-abortion provisions.

A few other Democrats voiced concern Friday about another issue, the bill’s Medicare funding formulas for doctors and hospitals. Liberals such as Reps. Peter DeFazio of Oregon and Michael Capuano of Massachusetts said they would withhold their support unless the formulas were rewritten.

But House Speaker Nancy Pelosi, D-Calif., exuded confidence, saying, “When we bring the bill to the floor, we will have a significant victory for the American people.”

The House is expected to vote Sunday on the health-care bill that the Senate approved on Christmas Eve, along with a separate package of amendments.

All 178 Republicans are expected to oppose the bill, so they need to peel off 38 Democrats to defeat the measure, almost the exact number that opposed the first version of the legislation in November.

The White House said that, just this week, the president has spoken 64 times to wavering lawmakers, often in one-on-one meetings in the Oval Office. That work paid off Friday when Reps. John Boccieri of Ohio, Allen Boyd of Florida, Suzanne Kosmas of Florida and Scott Murphy of New York announced their new support, bringing to seven the number of Democratic converts this week. Boccieri, Kosmas and Murphy are freshmen who Republicans have targeted in the November midterm elections.

If the bill is approved, Obama would sign the Senate version into law. The amendments to that law would be sent across the Capitol, where the Senate would try to approve them next week.

The House has already confronted the hurdle of abortion once in this year-long health-care debate.

Last November, antiabortion Democrats led by Rep. Bart Stupak of Michigan successfully pushed an amendment barring people who receive federal subsidies for insurance from using that money to buy policies covering abortions.

The Senate’s version included slightly less stringent restrictions; state-run insurance exchanges created under the legislation would be required to offer at least one policy that included abortion coverage, and recipients of federal tax credits for insurance would be permitted to buy it. Their tax credit would finance the bulk of their policy, but they would have to write a separate check, with their own money, to pay for the part of the policy that covers elective abortions.