Senators from both parties reached a deal Thursday to bring a bill that would all but ensure that the 9/11 Victim Compensation Fund never runs out of money to a vote next week, but squabbled over a GOP lawmaker's push for an amendment that would cap the budget allocating funds to first responders who survived the terrorist attacks.

Sens. Kirsten Gillibrand and Chuck Schumer, both D-N.Y., announced that they would vote against a proposal put forward by Sen. Mike Lee, R-Utah, that would allocate $10.2 billion to the fund over the next 10 years, which the Congressional Budget Office has estimated would be the result of approving a bill passed by the House last week. The current bill would authorize the fund through 2092, and lawmakers are unsure from where the money needed to keep the fund going for that amount of time will come.

"Since 2011, the 9/11 Victims Fund has always had finite authorizations, and by all accounts it has an excellent record avoiding waste and abuse," Lee said in a statement Thursday. "These two things are not coincidental. They go together."

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Schumer and Gillibrand, who called Lee's amendment "cynical," agreed to let the amendment go to a vote but said they were sure that it would not pass.

Sen. Rand Paul, R-Ky., who argued that the bill was "irresponsible" because of advocates' push for a sunset clause that would guarantee funding with no specific dollar amount for the next seven decades, told Fox News' "Your World with Neil Cavuto" Thursday that "we will vote and we will lose" but added that he was still willing to support the amendment out of concern for the federal deficit.

Gillibrand argued that 9/11 first responders and their advocates, which include comedian Jon Stewart, "shouldn't be having to come back to Washington again and again just to get the compensation that they earned and deserved."

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"The truth is the timing is limited for this bill because these men and women aren't going to survive," Gillibrand said on the floor of the Senate Thursday. "So many of them are already sick and dying and everything they care about is just being able to provide for their families. So there's nothing about this bill that's trying to play politics with the lives of men. There is going to be no fraud, no disuse. This is literally all that's necessary for these families to survive during these horrible times when their loved ones are dying."

The bill has 75 co-sponsors in the Senate, more than enough to override any potential presidential veto. The current $7.4 billion fund is expected to run dry by December 2020 and administrators recently cut benefit payments by up to 70 percent.

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More than 2,000 firefighters, police officers, federal agents, and other first responders have died from 9/11-related diseases and that number is expected to climb over time, according to the Centers for Disease Control and Prevention's World Trade Center Health Program.

"We need to stop these political games and pass this bill now," Gillibrand said in a statement on Thursday.

Fox News' Chad Pergram and The Associated Press contributed to this report.