"We are on the verge of a Christmas miracle,'' said Sen. Kirsten Gillibrand, D-N.Y.
Gillibrand and fellow New York Democratic Sen. Charles Schumer said the new version of the bill -- with a $6.2 billion price tag over 10 years, instead of $7.4 billion, and using an excise tax to pay for the legislation -- would likely win enough GOP support to pass.
Key Republican senators who Gillibrand and Schumer are courting include Susan Collins of Maine, Mark Kirk of Illinois and Lisa Murkowski of Alaska, The Wall Street Journal reported.
The bill needs support from two Republicans to pass.
Gillibrand and Schumer said they expected the Senate to consider the new bill after the upper house finishes dealing with the new Strategic Arms Reduction Treaty between Washington and Moscow.
"Barring a setback, we believe we are on the path to victory by the end of the week," Schumer said at the news conference Sunday.
Changing the bill to improve its chances in the Senate means the House will have to vote on the new version too.
A group of Sept. 11, 2001, responders plans to demonstrate in front of the White House Tuesday to urge U.S. President Barack Obama to support the revised bill.
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