WASHINGTON — Just a few unsolved problems — and one final sales job — stand between House Democratic leaders and a landmark vote on a remake of the nation’s health care system.

Unfortunately for the Democrats, the unresolved issues are among the most vexing out there: abortion and immigration.

Confident they’ll succeed, Democratic leaders have yet to nail down the votes they need to pass their sweeping bill. They’re aiming for floor action to begin as early as Friday and finish before Veterans Day, Nov. 11.

The 10-year, $1.2 trillion bill would extend coverage to tens of millions of uninsured Americans with a new requirement that almost everyone get insured, and give subsidies to help lower-income people afford plans. Businesses would be required to cover their employees, and insurance companies would no longer be able to turn people away because of their health status.

Democratic leaders unveiled the bill last week, a 1,990-page plan that sought to satisfy the concerns of both liberals and moderates while combining separate health bills passed by three different committees over the summer.

House leaders have made progress winning over liberals to back a new government insurance plan in the bill that’s weaker than what liberals preferred. But they haven’t quite closed the deal with enough members of their caucus. An initial vote tally showed them falling short of the 218-vote majority they need, according to Democratic aides.

“The bill’s improved,” said Rep. Jason Altmire, D-Pa.,  an undecided lawmaker, citing fewer taxes and more deficit-reducing measures. “I’m not sure it’s there yet.”

Across the Capitol, senators are waiting to see the final language and price tag on a health bill that Majority Leader Harry Reid and a few other top officials wrote in secret. It’s not clear when those details will be available and Reid, D-Nev., may not be able to begin debate on the issue until the week before Thanksgiving.

Determined to avoid being accused of not reading their bill, House Democrats met late Monday to do just that, going over the legislation section by section in a room in the basement of the Capitol.

Anti-abortion Democrats were circulating language to strengthen prohibitions in the bill against federal funding of abortion. The bill stipulates that people getting federal subsidies would have to use their own money to get abortions, but that division is not clear enough to satisfy some lawmakers. Lawmakers are considering language that would make a more straightforward declaration against use of federal funds for abortion.

On immigration, it’s still not settled whether illegal immigrants would be allowed to shop for insurance within a new purchasing exchange. Some lawmakers say that even if they use their own money to buy private plans they would be getting a benefit from the federally established exchange. The White House does not want illegal immigrants to access the exchange, and the Senate bill would keep them out.

After spending months criticizing Democratic health care plans, House Republicans are preparing to unveil one of their own, though Minority Leader John Boehner was short on details when he described it to reporters Monday.

He said it would omit one of the central provisions in Democratic bills — a ban on the insurance industry’s practice of denying coverage on the basis of pre-existing medical conditions. Instead, he said the Republicans would encourage creation of insurance pools for high-risk individuals and take other steps to ease their access to coverage.

Boehner also said Republicans would propose limits on medical malpractice lawsuits in what he said was an attempt to reduce the cost of coverage.

Despite their internal divisions, many Democrats are eager to vote for a health overhaul bill — even those who hadn’t quite committed themselves to doing so.

“We fail to pass health care reform at our own peril,” said Rep. Gerry Connolly, D-Va., though he has yet to agree to support the bill. “Our base expects health care reform. ... You can’t distance yourself from your base and expect to succeed.”

— AP Special Correspondent David Espo contributed to this report.