"Your World" host Neil Cavuto pushed back on several claims made by State Department spokesman Ned Price during a press conference on Monday.

Price claimed the Taliban wants to have "a relationship with the rest of the world" and be seen as a legitimate government rather than a global "pariah." He also alleged commitments of "safe passage" of Americans and U.S.-aligned individuals to the Hamid Karzai International Airport "don't have an expiration date."

However, Cavuto pointed to a Sky News interview with the Taliban's Doha, Qatar-based ombudsman, Suhail Shaheen—who contradicted Price and said an American presence after President Joe Biden's most recent deadline of August 31 cryptically represents a "red line."

"If [the US] extends it, that means we're extending occupation—there's no need for that. It will create mistrust between us," Shaheen said Monday.

Cavuto added Price's claim that the Taliban promised the aforementioned "safe passage" to the airport doesn't appear to be true.

"That doesn't appear to be the case right now," Cavuto said, as reports of Taliban beating Americans and Afghans trying to get to the airport along with house-to-house "hunts" by the militants for people who assisted U.S. operations in Afghanistan over the past two decades.

"He also said that we're acting with one voice: The fact of the matter is, the British, the Germans and the French have criticized our approach and inability or at least refusal to leave the perimeter that is now cordoned around the Kabul Airport that has been set up by the Taliban. So they're not speaking with one voice," Cavuto said.

Cavuto then turned to Fox News national security correspondent Jennifer Griffin at the Pentagon.

"The bottom line is the U.S. now needs the permission of the Taliban to stay," Griffin said. "The State Department is now saying American citizens, westerners with diplomats and SIV's with a Visa in hand should come to the Kabul airport and will be let through the gates – that will slow the flow to a trickle unless they waive other requirements and do the vetting in these third counties … Qatar, Germany and elsewhere."

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She continued, "For those that reached out to us that are in hiding that are telling us that the Taliban are knocking on their doors and they are cowering with their children and these are people that worked with the Americans for years, some of them didn't fill out SIV paperwork because they thought they would stay and work with the Afghan government: When that government collapsed, the whole system collapsed. These people do deserve to go because they're going to be killed if they're left behind."

Griffin added that U.S. military servicemembers on the ground have been trying to "push the system as much as possible" to get Americans and U.S.-aligned Afghans to safety.

"We've seen heroic efforts behind the scenes, but with the State Department process changing today to require vetting to be done on the ground at the airport, it's going to make it very difficult to keep that pipeline moving," she said.