With logistics and conditions at the Kabul international airport growing more dire, the Biden administration on Tuesday is scheduled to discuss with allies a withdrawal from Afghanistan beyond an Aug. 31 deadline that left just eight days to evacuate thousands of people.

Compounding the problem are the challenges borne of the more rapid than expected Taliban takeover. The State Department doesn’t know exactly how many Americans are left on the ground, although some have estimated that between 10,000 and 15,000 were there when the evacuations began, several U.S. officials said.

President Biden on Tuesday will meet virtually with the leaders of the Group of Seven leading nations to discuss coordination and evacuation plans, as well as the need for humanitarian assistance for Afghan refugees. On Sunday, Mr. Biden said he and his military advisers were discussing extending the Aug. 31 withdrawal deadline.

In response, the Taliban, through direct talks with the U.S. in Kabul, said it wouldn’t recognize any extension, nor would it assure that forces that stay in Afghanistan beyond Aug. 31 wouldn’t be subject to attack.

Mr. Biden’s national security adviser, Jake Sullivan, said Monday that the number of Americans on the ground is hard to assess because many Americans didn’t register with the embassy when they arrived in Afghanistan, and many didn’t deregister when they left. “Now, it’s our responsibility to find them,” Mr. Sullivan said.

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Getting them to the airport can be a treacherous exercise. Afghan, American and German soldiers were involved Monday morning in an exchange of fire at the airport that left one Afghan soldier dead and three wounded, German military officials said.

No injuries to American or German soldiers were reported. Some remnants of the Afghan army are working with international troops to provide security around the perimeter of the U.S.-controlled airport.

Beyond Americans, estimates from the government and private groups working on the evacuations suggest the number of Afghans seeking refuge is in the tens of thousands. Deciding who to accept, with the prospect of ultimately bringing them to the U.S., is a massive task for the Departments of State and Homeland Security.

Congressional offices had tallies of hundreds of American citizens they were advocating for, finagling flights and working tirelessly to get them past the Taliban checkpoints and into the airport.

“I continue to hear in real time first hand stories of people turned away by uniformed U.S. personnel even though they are on every list. And while new ways of getting people onto the base have begun to prove viable, it’s still extremely hard to get vulnerable Afghans in, and once there, conditions on the airfield are horrible,” said Rep. Tom Malinowski (D., N.J.).

Sen. Lindsey Graham (R., S.C.) told Fox News on Monday, “We’re going to leave a lot of people behind who fought for us so bravely.”

Following a classified briefing on the situation in Afghanistan Monday, House Intelligence Committee Chairman Adam Schiff (D., Calif.) said completing evacuations by the Aug. 31 withdrawal deadline is possible but “very unlikely.”

On Monday, about 10,900 people were evacuated from Kabul, a White House official said, through a combination of U.S. military cargo flights and coalition flights. Since Aug. 14, the U.S. has evacuated and facilitated the evacuation of about 48,000 people, the White House said.

According to a U.S. official involved in the effort, the U.S. is currently evacuating mostly U.S. citizens, green card holders, and the spouses and children of those individuals. Afghans who were employed by the embassy or other U.S. agencies in Afghanistan haven’t yet been evacuated in significant numbers.

“They’re getting very worried,” the official said. “These are the people we could have helped get out weeks ago.”

President Biden acknowledged the difficulty of getting thousands out of Afghanistan during a news conference Sunday. He also said U.S. airlines will help with evacuation efforts from staging centers in Europe and the Middle East. Photo: Andrew Caballero-Reynolds/AFP/Getty Images The Wall Street Journal Interactive Edition

“Unless we execute within a day or two, it’s not going to end well,” the U.S. official said, adding that the Taliban are currently screening documents on behalf of the U.S. outside the Kabul airport, and there is concern that the group may raise objections when it comes time to evacuate locally employed Afghan staff in larger numbers.

Many of the U.S. Embassy’s locally employed staffers weren’t moved to a secure location at the airport alongside American employees because they had been working outside the embassy compound “given the volatile security situation,” he said. Some Afghan staff members have since been relocated, State Department spokesman Ned Price said Monday.

Mr. Price added that the administration was working to expedite the screening process for Afghans, but cited the “rigorous vetting” required before admission to the U.S.

“That vetting, by and large, is taking place [in] third countries,” he said.

Some U.S. officials fear that if the U.S. stays beyond Aug. 31, security cooperation with the Taliban would collapse.

“We are well aware of the stated desire of the Taliban to have this mission completed by the 31st,” Pentagon spokesman John Kirby said Monday, adding that the goal is to complete the mission by then, and any need for an extension would be assessed in the coming days.

Mr. Sullivan said talks with the Taliban and others continue in the meantime. “Ultimately it will be the president’s decision how this proceeds, no one else’s,” he said. “We believe that we have time between now and the 31st to get out any American who wants to get out.”

Refugees from Afghanistan arrived at Dulles International Airport on Monday.

Photo: andrew caballero-reynolds/Agence France-Presse/Getty Images

A U.S. official said that the final evacuation was initially intended to ramp up around Aug. 27 since it takes several days to get thousands of troops, personnel and their equipment out of the country. That withdrawal initiation date on Monday was moved on to Aug. 29, further suggesting that withdrawal could stretch beyond Aug. 31.

U.S. and allied officials said that it is unclear to them whether the Taliban is bluffing in its threat, and that they would continue to negotiate directly with the Taliban and assess the security situation in the coming days.

Allies with the North Atlantic Treaty Organization, which fought alongside the U.S. in its war in Afghanistan and decided to pull out when Mr. Biden announced his intention to withdraw, are now considering the potential for their own extension since they too are struggling to withdraw their own nationals and the Afghans who worked alongside them.

Mr. Biden spoke on Monday with British Prime Minister Boris Johnson to coordinate evacuation efforts, the White House said. The British government said in its readout that the two leaders agreed to collaborate on evacuations “including after the initial phase of the evacuation has ended.”

French Foreign Minister Jean-Yves Le Drian said Monday that it is necessary to go beyond Aug. 31.

“We are concerned about the Aug. 31 deadline set by the United States. More time is needed to complete the current operations,” Mr. Le Drian said, according to a French pool reporter accompanying the minister to the United Arab Emirates.

His sentiments are increasingly being echoed in other European capitals as chaos outside the airport escalated.

Meanwhile, the U.S. was struggling to find new destinations for Afghan refugees as bases in Qatar, Germany and other locations quickly filled up and conditions there deteriorated.

To date, eight transit hubs in six countries “are hosting more than 17,000 evacuees and have beds available to accept more people,” an administration official said. Capacity is expected to increase as hubs open in additional countries; another 8,000 beds should become available within the next 24 hours, a State Department official said.

— Michael R. Gordon, Courtney McBride, Ben Kesling, Alex Leary and Siobhan Hughes contributed to this article.

Write to Vivian Salama at vivian.salama@wsj.com and Nancy A. Youssef at nancy.youssef@wsj.com

Corrections & Amplifications
An earlier version of this article misspelled Lindsey Graham’s name. (Corrected on Aug. 23, 2021.)