WASHINGTON – The most senior African-American in Congress offered some unsolicited advice Monday to President Barack Obama on how to get his signature health care bill through a balky Senate: knock their heads together.

“The president could take a few pages from Lyndon Johnson’s book … and start knocking heads together,” said Rep. John Conyers, D-Mich., chairman of the House Judiciary Committee.

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Conyers, who spoke to reporters in Detroit, first came to Congress in 1965, the year Medicare and the Voting Rights Act both passed under the strong hand of Johnson, by then the president. Obama was not yet 4 years old.

The House passed the sweeping health care legislation late Saturday on a narrow 220-215 vote. By Sunday, the bill was being declared a nonstarter in the Senate.

A government health insurance plan included in the House bill is unacceptable to a few Democratic moderates who hold the balance of power in the there. These senators are locked in a battle with liberals, with the fate of health care legislation at stake.

On Sunday, Obama urged the Senate to follow the lead of the House, like a runner in a relay race. The president says he’s confident senators will do the right thing.

But Sen. Joe Lieberman wasted no time in saying he can’t support the public insurance plan in the House bill.

If a government plan is part of the deal, “as a matter of conscience, I will not allow this bill to come to a final vote,” said Sen. Joe Lieberman, the Connecticut independent whose vote Democrats need to overcome GOP filibusters.

“The House bill is dead on arrival in the Senate,” Sen. Lindsey Graham, R-S.C., said.

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Democrats did not line up to challenge him. Senate Majority Leader Harry Reid, D-Nev., has yet to schedule floor debate and hinted last week that senators may not be able to finish health care this year.

Nonetheless, the House vote provided an important lesson in how to succeed with less-than-perfect party unity, and one that Senate Democrats may be able to adapt. House Democrats overcame their own divisions and broke an impasse that threatened the bill after liberals grudgingly accepted tougher restrictions on abortion funding, as abortion opponents demanded.

In the Senate, the stumbling block is the idea of the government competing with private insurers. Liberals may have to swallow hard and accept a deal without a public plan to keep the legislation alive. As in the House, the compromise appears to be to the right of the political spectrum.

Republican Sen. Olympia Snowe of Maine, who voted for a version of the Senate bill in committee, has given the Democrats a possible way out. She’s proposing to allow a government plan, if after a few years premiums keep escalating and local health insurance markets remain in the grip of a few big companies. This is the “trigger” option.

That approach appeals to moderates such as Sen. Mary Landrieu, D-La. “If the private market fails to reform, there would be a fallback position,” Landrieu said last week. “It should be triggered by choice and affordability, not by political whim.”

Lieberman said he opposes the public plan because it could become a huge and costly entitlement program.

For now, Reid is trying to find the votes for a different approach: a government plan that states could opt out of.

He will keep meeting with senators this week to see if he can work out a political formula that will give him not only the 60 votes needed to begin debate, but the 60 needed to shut off discussion and bring the bill to a final vote.

Toward the end of the week, the Congressional Budget Office may report back with a costs-and-coverage estimate on Reid’s bill, which he assembled from legislation passed by the Finance Committee and the Health, Education, Labor and Pensions Committee. The Finance Committee version does not include a government plan.

Reid has pledged to Obama that he will get the bill done by the end of the year and remains committed to doing that, according to a Senate leadership aide.

Both the House and Senate bills gradually would extend coverage to nearly all Americans by providing government subsidies to help pay premiums. The measures would bar insurers’ practices such as charging more to those in poor health or denying them coverage altogether.

All Americans would be required to carry health insurance, either through an employer, a government plan or by purchasing it on their own.

The majority of middle-class workers with employer-provided health insurance would not see major changes. The main beneficiaries would be some 30 million people who have no coverage at work or have to buy it on their own. The legislation would create a federally regulated marketplace where they could shop for coverage.

Lieberman appeared on “Fox News Sunday,” while Graham was on CBS’ “Face the Nation.”

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