Updated 10:18 am Eastern Time
New York Democratic Sen. Kirsten Gillibrand said on CBS' "The Early Show" Monday morning that a bill to provide health care for first responders to the Sept. 11 terrorist attacks has the votes to pass thanks in part to a decision to reduce the cost of the bill from $7.4 billion to $6.2 billion.
She said "we have the support we need" to win passage immediately following a planned Senate vote on the New Start treaty.
"We've been working very hard for many months and over the last several weeks on really addressing some of the concerns that my Republican colleagues had about the bill," Gillibrand told "Early Show" co-anchor Jeff Glor. "That's why we changed how we are paying for the bill. We've also changed the total amount of the bill to reflect the fact that we've just had a settlement and some more funds are getting to these first responders. I believe we have the package we need to really have a Christmas miracle, what we're aiming for."
Gillbrand said a bill to cover the health care of first responders is essential to provide care to "the men and women that were racing up the towers when everybody was coming down, and they're the ones who tried to find survivors and then remained."
"These are the men who have sacrificed so much," she said. "And I can tell you a lot of them are dying right now, dying of terrible cancers and other diseases and it's heartbreaking because these men are young, these men and women are very young."
Gillibrand pointed to 29 New York City police officers who have died because of diseases picked up in the dust at the twin towers site, and said their average age was just 46.
"So, these are men and women that are being cut down in the prime of their lives because of their heroism, because they responded when duty called," she told Glor.
Researchers and first responders say exposure to the twin towers site immediately following the attacks is responsible for asthma problems and other illnesses, though some have questioned to what degree the exposure can be blamed for such health problems.
Appearing on ABC's "Good Morning America," New York's other Democratic senator, Charles Schumer, also said the votes were there for passage -- but said he was concerned that delays in the waning days of the lame-duck session could still kill the bill. The Senate needs to deal with the New Start nuclear arms treaty with Russia first -- a vote could take place Wednesday -- and the necessary series of votes on the 9/11 bill might not be possible before members of Congress go home for Christmas.
"And that's my plea to my colleagues in both the House and Senate, please don't delay this bill. Let it come to a vote and we will win," he said. Schumer and Gillibrand have pushed hard for the bill and introduced the modified version Sunday
Both the House and Senate must approve the bill before the end of the year. If they do not, the matter gets kicked to the new, more Republican Congress in January, where prospects for passage are slimmer.
House Speaker Nancy Pelosi is vowing to do everything possible to get the new version of the bill passed in her chamber before the lame-duck session ends, though it is not clear there there will be time to do so. (The House passed the $7.4 billion version of the bill in September.) Over the weekend, Schumer urged the House to stay in session to pass the bill.
The bill got 57 votes in a recent Senate test vote, three short of the 60 it needs to break a Republican-led filibuster. One of those no votes, however, was Senate Majority Leader Harry Reid, who switched his vote at the end so that he could bring the bill up again. That means only two votes need to be picked up for passage, assuming no senator switches from yes to no.
The new version of the bill addresses Republican objections that it amounted to a corporate tax increase, the Associated Press reports. They objected to a requirement that multinational corporations pay taxes on income earned in the United States even if they are incorporated in a tax haven.
The new version pays for the costs of the bill with a fee on some foreign companies that get procurement contracts from the U.S. government. It also calls for an extension of some visa-related fees.
Gillibrand also predicted passages of the New Start treaty during her appearance on "The Early Show." Check out the interview above.
Brian Montopoli is senior political reporter for CBSNews.com. You can read more of his posts here. Follow Hotsheet on Facebook and Twitter.