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Fed-up Republican lawmakers are coming around to the possibility of President Trump issuing an executive order in place of "Phase 4" coronavirus relief legislation, since Congress is making "no progress" in negotiations, Sen. John Kennedy, R-La., told "The Story" Tuesday.

"The reason we've made no progress," Kennedy told host Sandra Smith, "is because Senator [Chuck] Schumer and Speaker [Nancy] Pelosi's position ... is Speaker Pelosi's $3 trillion HEROES act, take it or leave it."

He added: "You don't have to be Einstein's cousin to understand that Speaker Pelosi's bill is not going to pass the United States Senate, nor should it."

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Lawmakers began negotiating the terms of the newest stimulus package last week hoping to reach an agreement before enhanced unemployment benefits, which provided an additional $600 per week to those without work, expired last Friday.

Democrats want the next package to continue the $600 weekly payments, but Republicans are concerned about Americans collecting more money than they earned from their jobs, thus becoming disincentivized to return to work.

The Republican HEALS Act, which bears a $1 trillion price tag, provides an extra $200 per week in unemployment insurance.

Kennedy said Pelosi's HEROES act "would permanently change ... our election laws, our immigration laws, and would offer free abortions. It would let prisoners go, it would bail out every mismanaged state pension plan, and she knows, and Senator Schumer knows, that that bill is not going to pass."

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Schumer, D-N.Y., slammed Republicans Tuesday, accusing them of "dragging their feet" and engaging in a political stunt.

"I would say at this juncture and with respect to this particular moment, Senator Schumer is very well-versed in the stuff that he's making up," said Kennedy, who added that Senate Republicans plan to propose another bill on Wednesday that would extend unemployment compensation benefits, and urged his "Democratic friends" to vote for it.

When asked about the possibility of Trump taking matters into his own hands, Kennedy replied: "Normally, I would say 'no' because this is Congress' job, but if we are not going to do our job and given the fact that the American people, through no fault of their own, find themselves in a pool of misery, I think the president would have to act."

A senior administration official told Fox News Monday that Trump is considering taking "a number of options" that would not require congressional approval, including the use of his controversial executive order privilege.

"Now, I’d want to see what he is proposing to do," Kennedy said. arguing that Trump's push for a payroll tax cut or payroll tax holiday "wouldn't help the people who through no fault of their own don't have a job."

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When Smith asked Kennedy if he could provide an approximate timeline for the arrival of unemployment relief, the senator said, "all I can do is look you in the eye and look the American people in their eyes and tell them the truth, that we are not making any progress."

"It's impossible to make progress when the only position on the table is, 'Pass Speaker Pelosi's bill or nothing,'" he said. "Those aren't reasonable alternatives, and that seems to be where we find ourselves. I'm beginning to believe that some are more worried about the next election instead of the next generation."

Fox News' Brooke Singman contributed to this report.