Overnight, nine American C-17 transport aircraft arrived at the airport delivering equipment and roughly 1,000 troops, Taylor said. Separately, seven C-17s departed the airport carrying a total of roughly 700-800 passengers. Of that number, 165 were American citizens and the remainder were a combination of Special Immigrant Visa applicants and third country nationals.

Taylor also referenced a photo published Monday by Defense One showing roughly 640 Afghans and their families crowded inside a C-17, being lifted to safety. The image, he said, “speaks to the humanity of our troops in this mission. The skill and professionalism of our U.S. military.”

Over the next 24 hours, “the speed of evacuation will pick up,” Taylor said, with the U.S. military expected to achieve roughly “one aircraft per hour in and out” of the airport in Kabul.

“We predict that our best effort could look like 5,000 to 9,000 passengers departing per day,” he said. “But we are mindful that a number of factors influence this effort, and circumstances could change.”

While the airport itself remains secure, the situation just outside is deteriorating rapidly. Crowds of desperate Afghans are rushing the outer perimeter, which is guarded by Afghan forces, according to three people with knowledge of the situation. Multiple Afghan interpreters could not get through to the airport to catch flights out of the country because of the chaos, the people said.

Earlier in the day, gunfire broke out in the area, two of the people said. Afghan guards fired into the air and threw flash bang grenades in an attempt to disperse the crowds swarming the perimeter, one of the people said.

The Taliban’s control of Kabul and checkpoints outside the airport also are reportedly complicating U.S. evacuation efforts, with time running out to remove vulnerable Americans and Afghans ahead of Biden’s self-imposed Aug. 31 deadline for the military mission.

Taylor said Tuesday the U.S. military has “had no hostile interactions, no attack and no threat by the Taliban” thus far at the airport, adding: “We remain vigilant.”

But Pentagon spokesperson John Kirby was vague when asked at the news briefing whether the United States had reached an agreement with the Taliban to allow the U.S. evacuations to proceed until the end of the month.

“Our commanders at the airport are in communication with Taliban commanders on the ground outside the airport. There have been discussions. There is communication between them and us. And I would just let the results speak for themselves,” Kirby said.

“Right now, as the general made clear, the mission runs through Aug. 31,” he added. “The commander in chief made it very clear that we were to complete this drawdown by Aug. 31, which now includes … the pulling out of American citizens and drawdown of our embassy personnel. So that’s what we’re focused on. That’s the timeline we’re on.”

Kirby also did not elaborate upon potential U.S. efforts to ease passage through Taliban checkpoints or expand the U.S. military perimeter around the airport, so Americans and Afghans could more easily reach their evacuation flights.

“There are interactions down at the local level,” he said. “And as the general said, we are processing American citizens to get out. So again … without speaking to the sausage-making of communications here, thus far — and it’s early on — the results are speaking for themselves.”

National Security Adviser Jake Sullivan spoke more definitively at a White House news briefing Tuesday afternoon, telling reporters the Taliban “have informed us that they are prepared to provide the safe passage of civilians to the airport, and we intend to hold them to that commitment.”

Pressed on the deadline for the apparent deal with the Taliban on airport access, Sullivan said U.S. officials “believe that this can go until the 31st.”

“We are talking to them about what the exact timetable is for how this will all play out, and I don’t want to negotiate in public on working out the best modality to get the most people out in the most efficient way possible,” he said.

Sullivan also downplayed concerns about Americans and Afghans safely making their way to the U.S. evacuation flights, saying: “By and large, what we have found is that people have been able to get to the airport.”

“In fact, very large numbers of people have been able to get to the airport and present themselves,” Sullivan added. “There have been instances where we have received reports of people being turned away or pushed back or even beaten. We are taking that up in a channel with the Taliban to try to resolve those issues. And we are concerned about whether that will continue to unfold in the coming days.”

State Department spokesperson Ned Price similarly was pressed Tuesday about reports of people unable to make it safely to the airport. Even if an individual has been notified and given instructions about evacuating, “if they feel that it is unsafe for them to make their way to the airport, they should not seek to do so,” he told reporters.

“We will continue to do all we can, too. And we will continue to be in touch with them, I should say, to provide clear guidance about when and how they should make their way to the airport compound,” Price said.

Price also was asked about plans for the remaining embassy staff in Kabul once the relocation efforts conclude.

“We’re focused on the mission at hand. … That is an effort to relocate, in some cases repatriate to the United States, in other cases relocate to third countries, as many individuals as we can, over as much time as we might have,” Price said. “Right now, we are thinking about this in terms of Aug. 31. If it is safe and responsible for us to potentially stay longer, that is something that we may be able to look at.”

Price did not say whether he was referring to the entire remaining diplomatic presence on the ground, but said the the department’s “first responsibility” is to the “safety and security” of that team.

The remarks from the senior administration officials came hours after the United States resumed operations Monday at the airport following efforts by U.S., Turkish and other international troops to reestablish security there. Thousands of desperate Afghans had stormed the tarmac in a series of overnight breaches from the airfield’s civilian, southern side — seeking to flee their country after the Taliban’s government takeover.

The pandemonium resulted in the United States suspending flights out of Kabul amid a rapidly deteriorating security situation and an urgent U.S. military operation to evacuate American civilians and Afghan allies out of the Taliban-controlled capital.

On Tuesday morning, however, Kirby rejected the notion that the administration was caught flat-footed by the disorder at the airport, explaining that U.S. officials has been “planning for noncombatant evacuation operations” since May — “right after” Biden announced his withdrawal decision in April.

“In fact, we held a big drill here at the Pentagon, downstairs in the Joint Operations Center with the entire interagency, to walk through what the retrograde was going to look like — the withdrawal — as well as including the possibility for these kinds of evacuation operations,” Kirby told MSNBC in an interview.

Kirby also described another drill that took place “just as recently as two weeks ago,” when U.S. officials “held a tabletop exercise here at the Pentagon to walk through what it would look like to do exactly what we’re doing now — which is a noncombatant evacuation operation from Hamid Karzai International Airport.”

Despite that planning, at least seven people died Monday during the storming of the airport in Kabul, including several Afghans who clung to a departing U.S. military jet and fell mid-air as it gained altitude. Additionally, the body of one Afghan was found in the landing gear of a C-17 hours after it hastily took off from the runway.

U.S. troops also were twice forced to respond to “hostile threats” at the airport, resulting in the deaths of two armed individuals who were shooting at them, as well as the possible wounding of an American soldier. No such security incidents have taken place since, Taylor said.

Kirby insisted Tuesday that U.S. officials had “planned for almost every contingency” surrounding evacuations. “But as an old military maxim says, no plan survives first contact. So obviously, we had to adjust in the moment,” he said.

“It would have been difficult to predict for the level of mayhem and chaos that we saw there,” Kirby added. “We’re mindful of the images, the graphic nature of them. Certainly, nobody wanted to see it result like it did over the last 24 hours.”

While the military, northern side of the airfield “is back up and running again,” Kirby said U.S. officials would continue to better secure the civilian, southern side throughout Tuesday.

“No plan is ever perfect, and no plan can be perfectly predictive in terms of what friction, what unknown aspects and factors you’re going to deal with on the backside,” he said.

Lara Seligman, Alex Ward and Myah Ward contributed to this report.