(CNN) -- As a vote on health care nears, House Democratic leaders are trying to woo party holdouts still wrestling with their decision.
An ongoing CNN analysis shows that opposition in the House to the Senate health care plan has reached 204 members -- 12 votes shy of defeating it.
Rep. Jason Altmire, D-Pennsylvania, said Wednesday that while he thinks the current version of the bill is better than what the House voted on in November, he's not ready to throw his support behind it.
Altmire told CNN's "American Morning" that he first needs to see the finished product and the estimated price tag from the nonpartisan Congressional Budget Office.
The House is expected to vote this week on the roughly $875 billion bill passed by the Senate in December. After the bill is signed into law, the House would then vote on a package of "fixes" proposed by President Obama.
Altmire echoed other House Democrats who are skeptical of the Senate bill, saying he wanted to strike the provision that exempts Nebraska from paying increased Medicaid expenses.
"The only absolute, definitive red line that there is for me is I will not vote for a bill that increases the deficit by even one penny," he said.
Some wavering House Democrats have also expressed concern the Senate bill does not include an adequate level of subsidies to help middle- and lower-income families purchase coverage. They also object to the Senate's proposed tax on high-end insurance plans.
Freshman Rep. John Boccieri, D-Ohio, said he is faced with "voting on an imperfect bill or doing nothing."
"Whether I serve two terms or one, I just want to make the right decision for the people of our district and Ohio and the country," he said.
Another Ohio Democrat, Rep. Dennis Kucinich, announced Wednesday that he plans to vote in favor of the Senate bill. Kucinich voted against the House bill in November.
"I have doubts about the bill. I do not think it is a step toward anything I've supported in the past," Kucinich said. "However, after careful discussions with President Obama, Speaker Pelosi, my wife Elizabeth and close friends, I've decided to cast a vote in favor of the legislation."
Two top Republican vote-counters, Senate Minority Whip Jon Kyl and House Minority Whip Eric Cantor, released a memo last week saying that a relatively small number of Democrats hold a tremendous amount of sway on the issue.
"We believe House passage of the Senate's health care bill will ultimately be decided by the 37 remaining House Democrats who voted NO to a government takeover last November, and the ... 21 House Democrats who originally voted YES, but may now be on the fence," they wrote.
House Speaker Nancy Pelosi may call for a vote on a rule that would simply "deem" the Senate bill to be passed instead of making each lawmaker voice support or opposition. The House then would proceed to the separate vote on the more popular changes to the Senate bill.
The idea has been dubbed the "Slaughter" solution, named after New York Democratic Rep. Louise Slaughter, who is the chair of the House Rules Committee.
Congress first used the self-executing rule in 1933, according to a memo sent to reporters by Vince Morris, spokesman for the House Rules Committee.
"It's been used often. It's been used by both parties. There's nothing unconstitutional about it. It's perfectly legitimate," Slaughter told CNN.
Slaughter said she sees "no necessity" in having to take a vote on a bill that includes provisions House Democrats don't support.
House Minority Leader John Boehner, R-Ohio, said Tuesday that Republicans will try to block the procedure. They will try to force a vote on a resolution requiring the Senate health care bill to be brought to an up-or-down vote.
Pelosi told reporters she wouldn't make a firm decision on strategy until receiving a final cost estimate on the legislation from the CBO.
Altmire said he doesn't support Slaughter's idea because it "increases the opportunity for the public to say, 'You know what, I'm not comfortable with this process.'"
"The polls at best are mixed on this, perhaps even tilting toward public reluctance to do this. I think the biggest thing is that the public has to accept the process and policy that we've passed," he said.
CNN's Dana Bash, Kristi Keck and Alan Silverleib contributed to this report.