WASHINGTON — Under White House pressure to act swiftly, House and Senate Democratic leaders reached for agreement Friday on President Barack Obama’s health care bill, sweetened suddenly by fresh billions for student aid and a sense that breakthroughs are at hand.

“It won’t be long,” before lawmakers vote, predicted Speaker Nancy Pelosi. She said neither liberals’ disappointment over the lack of a government health care option nor a traditional mistrust of the Senate would prevent passage in the House.

At the White House, officials worked to maximize Obama’s influence over lawmakers who control the fate of legislation that has spawned a yearlong struggle. They said he would delay his departure on an Asian trip for three days — until March 21 — and he will go to Ohio next week for a campaign-style pitch for his health care proposals.

The delay gives congressional leaders much-needed breathing room to finish the legislation and nail down support from wavering lawmakers.

“I’m delighted that the president will be here for the passage of the bill; it’s going to be historic,” said Pelosi, D-Calif. — though there’s no guarantee the House can act by then. A procedural vote in the House Budget Committee is set for Monday afternoon, but as of late Friday lawmakers still hadn’t gotten the final analysis from the Congressional Budget Office that they need to go forward.

With Democrats deciding to incorporate changes in student aid into the bill, Republicans suddenly had a new reason to oppose legislation they have long sought to scuttle.

“Well of course it’s a very bad idea,” said Senate Republican leader Mitch McConnell of Kentucky. “We now have the government running banks, insurance companies, car companies, and they do want to take over the student loan business.”

He said it was symptomatic of Democrats’ determination to have “the government expand its tentacles into absolutely everything.”

At its core, the health care bill is designed to provide health care to tens of millions who lack it and ban insurance companies from denying medical coverage on the basis of pre-existing medical conditions. Obama also wants the measure to begin to slow the rate of growth in medical costs nationwide. Most people would have to get insurance by law, and poor and middle-income Americans — including families of four earning up to $88,000 — would receive subsidies.

Whatever the outcome, there was no doubt the issue would reverberate into this fall’s elections, with control of Congress at stake.

The health care bill appeared on the cusp of passage in early January, but was derailed when Senate Republicans won a Senate seat in Massachusetts, and with it, the strength needed to sustain a filibuster and block a final vote.

In the weeks since, the White House and Democrats have embarked on a two-part rescue strategy. It calls for the House to pass legislation that cleared the Senate in December, despite numerous objections, and for both houses to follow immediately with a second bill that makes changes to the first.

The second, fix-it bill would be drafted under rules that strip Senate Republicans of the ability to require Democrats produce a 60-vote majority.

Obama outlined numerous requested changes several weeks ago, many of them designed to satisfy the concerns of House Democrats.

They would increase subsidies for lower income families who cannot afford insurance, give additional money to states that provide higher-than-average benefits under Medicaid, and gradually close a coverage gap in the Medicare prescription drug program used by millions of seniors.

Congressional officials said all three issues would be addressed in the fix-it bill, although other administration requests remained in doubt. The president wants creation of a commission with authority to force savings in Medicare, for example, and is seeking the deletion of items sought by individual senators.

Those were among the issues still in dispute after days of secretive talks involving the White House and House and Senate leaders.

The decision to add far-reaching student aid changes to the bill had its roots in obscure parliamentary rules governing the Senate’s debate of the legislation. But House Democrats and the White House quickly seized on it as a way to advance a top administration priority that lacks the 60 votes needed to clear the Senate otherwise.

The measure would require the government to originate student loans, closing out a role for banks and other private lenders who charge a fee. Obama proposed taking the savings and plowing it into higher Pell Grants that go to needy college students.

Officials said that under current estimates, the change would free as much as $66 billion over a decade, although Pelosi indicated she wanted it spread beyond Pell Grants to other education programs.

At a news conference, the speaker confessed to being disappointed that the legislation would not include a government-run health care option, but said other parts of the legislation would hold insurance companies accountable.

The tussle over a public option roiled Democrats for months, but has subsided in recent weeks.

“We’ve crossed that bridge,” said Rep. Bill Pascrell, D-N.J. “Those people who were saying ’public option’ are muted right now. That’s done. It’s not going to happen.”

At a closed-door meeting of the rank-and-file, House Democratic leaders sought to allay concerns that Senate Democrats might simply refuse to pass the fix-it bill after the House swallows the measure it doesn’t like.

Rep. G.K. Butterfield, D-N.C., said party leaders told the House caucus they have “a firm commitment” from the Senate to do its part.