Stupak and some other Democrats have been firm about their abortion language but say they are open to working with Senators to craft a compromise.
Another thorny issue is whether the health care bill should include the option of a government-run insurance plan that would compete with private insurers. In August, 60 Democrats signed a bill saying they would only vote for a bill if it included a public option, but much to the chagrin of some, the Senate ended up taking it out to appease lawmakers such as Nelson and Independent Sen. Joe Lieberman of Connecticut.
Lawmakers are also closely watching the results of the Massachusetts senate seat formerly occupied by late Sen. Ted Kennedy. On Jan. 19, there is a special election to fill the seat, and if Republican candidate Scott Brown prevails, Democrats would no longer have the 60 votes they need to pass the bill in the Senate, making the final passage even tougher.
Despite these issues, Democrats are confident they can come together on the health care bill, but the negotiations between the two chambers of Congress will be tough with little or no room for error.
On Wednesday, Speaker of the House Nancy Pelosi, D-Calif., was at a loss for words when asked if a final bill would pass by the end of January.
"It's possible, it's possible," Pelosi said as she and chairs of key House committees emerged from a two-hour meeting with President Obama at the White House. "We will bring the bill to the floor when we are ready, and hopefully that will be soon."
Democratic leaders are confident that they will ultimately prevail.
Pelosi said that it has been a "very intense couple of days" as House, Senate and White House staffers engage with each other to review and suggest changes to the language of the legislation.
"I think we are very close to reconciliation respectful of the challenges, policy and otherwise in the House and Senate," Pelosi said, but she wouldn't reveal the details of their meeting with Obama, except to say the work done was "very productive."
The White House said one of the issues the president discussed with lawmakers was how to pay for health care overhaul. As for the question of whether he pushed them to take the fast track approach, White House Press Secretary Robert Gibbs would only say, "The president wants to get a bill to his desk as quickly as possible."
ABC News' Jonathan Karl and Sunlen Miller contributed to this report.