(CNN) A 9/11 first responder is "satisfied" with a commitment from Senate Majority Leader Mitch McConnell to hold a vote to extend the compensation fund for victims of the attack, but he's still extremely frustrated with Congress, saying, "We pay for that shit."

"The chairs that they put their asses in, the pens that they use, the pads that they write on, we pay for that shit," 9/11 first responder John Feal told reporters after meeting with McConnell. "That's us. They work for us. Mitch McConnell works for us. He works for all of you guys."

"Today, Mitch McConnell promised to work for us," he added. "I'm going to take him for his word."

According to Feal, the Kentucky Republican committed on Tuesday to holding a vote to extend the September 11th Victim Compensation Fund, after sitting down with Feal and other 9/11 first responders on Capitol Hill. The meeting came after comedian Jon Stewart's high-profile criticism of Congress and his plea for the extension of the September 11th Victim Compensation Fund , which is struggling to pay its current claims.

"Mitch McConnell made a commitment to the 9/11 community and my team leaders that he is going to help us get a piece of legislation that is going to be passed in the House in July, for an August vote in the Senate," Feal said. "To get his commitment today and all of them in the room heard it -- we are satisfied."

McConnell's office did not immediately respond to a request for comment on the commitment.

Feal told CNN the night before the meeting that it had been in the works since before Stewart's emotional testimony on Capitol Hill two weeks ago, but plans had not formally come together until that week that the comedian and activist's impassioned plea went viral

At issue is funding for the September 11th Victim Compensation Fund, which provides health care and services for 9/11 first responders. The current law, which was last renewed in 2015, expires next year and the fund's administrator says it doesn't have enough money to pay out all the current claims now. The new bill, that passed out of the House committee last week, does not call for a specific amount of funds but whatever sums necessary through 2090.

At the time of its last renewal in 2015, Congress appropriated $4.6 billion to the fund, bringing the total appropriated amount of the fund over the years to $7.4 billion. The special master who administers the fund anticipates that total payouts for claims filed before the measure expires in 2020 could be far higher: $11.6 billion, if a current uptick in claims -- largely caused by an increase in serious illnesses and deaths -- continues.

While there have been public commitments from both parties to ensure more money the fund is extended, the legislation seems at least weeks if not months away from final passage -- a timeline that is too slow for some of the funds' recipients and their advocates.

One example: a retired NYPD bomb squad detective who testified about his 9/11-related medical issues alongside Stewart said last week he is now entering end-of-life hospice care.

"I'm now in hospice, because (there) is nothing else the doctors can do to fight the cancer," Luis Alvarez wrote in a Facebook post last week.

In the meeting the group first responders gave McConnell Alvarez's badge.

"We wanted to Senate Majority Leader to be reminded of people like Detective Luiz Alvares," Feal said after the meeting. "So he's got his badge now and if he strays from his commitment then we will go back into attack mode. But for now, we are going to put down all swords."

The first responders said they are still expecting the bill to be passed as a standalone bill, not attached to another bill, but said that McConnell did not make the promise to do so.

Stewart, who has been fighting for 9/11 first responders' health benefits for years, testified on Capitol Hill earlier this month at a House subcommittee hearing over the legislation to fight for the funding to be extended immediately and he called out lawmakers for not attending the hearing.

"'Shameful," Stewart said, "It's an embarrassment to the country and it is a stain on this institution. And you should be ashamed of yourselves for those that aren't here. But you won't be, because accountability doesn't appear to be something that occurs in this chamber."

Following Stewart's comments, lawmakers from both parties have vowed to extend the funding for the program.

"Every sick responder and survivor should be treated with the same dignity and compassion," said House Judiciary Chairman Jerry Nadler, a Democrat from New York whose committee unanimously advanced the legislation. "All responders and survivors, whether they got sick in 2015 or will get sick in 2025 or 2035, should be properly compensated. Congress must act to make that happen."

In the days since his initial comments, Stewart has kept up the public pressure, focusing much of his criticism on McConnell as well as challenging him to meet with first responders as soon as possible, something McConnell has done before.

"Don't make them beg for it. You could pass this thing as a standalone bill tomorrow," Stewart said appearing on CBS' "The Late Show with Stephen Colbert." "If you're busy, I get it. Just understand, the next time we have war; or you're being robbed; or your house is on fire, and you make that desperate call for help, don't get bent out of shape if they show up at the last minute, with fewer people than you thought were going to pay attention, and don't actually put it out."

McConnell pushed back in a Fox News interview, vowing to the fund would be fully funded and saying he could not understand why Stewart is the one "all bent out of shape" on the issue.

"Well, many things in Congress happen at the last minute. We never failed to address this issue and we will address it again," McConnell told Fox News. "I don't know why he is all bent out of shape. We will take care of the 9/11 compensation fund."

McConnell dodged the question last week when asked by CNN if he would meet with 9/11 responders in response to Stewart's challenge, saying only, "I don't know how many times I can say we've never let 9/11 victims behind and we won't again."

This was not be McConnell's first meeting with a group of first responders. Feal tells CNN they last met with McConnell in 2015 and 2010. McConnell's office did not comment, when asked by CNN about Tuesday's meeting.

Feal said McConnell used the word "urgency" in the meeting Tuesday and described this meeting as different than previous ones.

"He actually sat for this one, the other ones he was quick to get up and leave his staff with us," Feal said, "You are going to have to ask the Senate majority leader -- was it Jon Stewart? Was it Luiz Alvarez or Lt. Mike O'Connell testimony in the Judiciary hearing?"

The bill now waits to be passed by the full House. House Majority Leader Steny Hoyer said it is their intention to take it up sometime before the August recess.

After passing the House, which it is expected to do easily, the bill will be sent to the Senate for a vote.

Senate Minority Leader Chuck Schumer, a New York Democrat, said last week he believes that Stewart's public pressure has "had some effect" on pressuring McConnell.

"By the end of Senator Mitch McConnell's meeting with these first responders tomorrow, he should commit to put the bill on the floor for a stand-alone vote immediately following House passage," Schumer said in a statement Monday. "We cannot tolerate any more delay."

This story has been updated with additional developments Tuesday.