DOHA, Qatar — With the massive United States military effort to evacuate American citizens and endangered Afghans completed, those still scrambling to find safe passage from Afghanistan are now navigating a complicated and potentially dangerous diplomatic impasse.

Unable to fly from Kabul’s airport, which remains closed and in need of upgrades, many people have flocked to the airport in the northern city of Mazar-i-Sharif, making it the latest flash point as the United States struggles to coordinate with its former Taliban adversaries to help people wanting to leave the country.

Secretary of State Antony J. Blinken, speaking to reporters during a visit to Qatar on Tuesday, said U.S. officials were “working around the clock” to ensure that flights carrying Americans and endangered Afghans can safely depart Afghanistan. He contested claims that the Taliban have blocked charter flights from departing the Mazar-i-Sharif airport.

Mr. Blinken said he was unaware of any “hostage-like” situation in Mazar-i-Sharif, contradicting a claim by a prominent House Republican that the Taliban were reneging on promises they have made to U.S. officials to allow safe passage from the country of foreigners and Afghans with valid travel documents.

“We have been assured, again, that all American citizens and Afghan citizens with valid travel documents will be allowed to leave,” Mr. Blinken said, adding that “we intend to hold the Taliban to that.”

Mr. Blinken, who appeared alongside Secretary of Defense Lloyd J. Austin III and their Qatari counterparts, said Taliban leaders had recently reaffirmed that commitment. He pointed to the exit from the country on Monday by an American family who used an undisclosed land route. The Taliban knew what the family was doing but did not impede them, U.S. officials say.

But in the case of Mazar-i-Sharif, Mr. Blinken said, the Taliban have objected to charter flights that combine passengers who have valid travel documents and those who do not.

“It’s my understanding that the Taliban has not denied exit to anyone holding a valid document, but they have said that those without valid documents at this point can’t leave,” Mr. Blinken said. “But because all of these people are grouped together, that’s meant that flights have not been allowed to go.”

On Sunday, Representative Mike McCaul of Texas, the top Republican on the House Foreign Relations Committee, said on “Fox News Sunday” that the Taliban were blocking the departure of six flights from Mazar-i-Sharif that included American citizens and Afghans who had acted as interpreters for the U.S. military. Mr. McCaul said the Taliban were holding the passengers “hostage” as they made demands of the U.S. government.

Mr. Blinken said he believed around 100 American citizens remain in Afghanistan, including “a relatively small number” seeking to leave Mazar-i-Sharif.

Further complicating matters, the Taliban said Tuesday they would not allow people to leave the country until a new government is formed. The Taliban announced a list of people who will fill key roles on Tuesday afternoon, but they held off on formally swearing in the new government.

The group said earlier in the day that without functioning ministries to grant exit stamps and perform other necessary duties, an orderly departure process was not yet in place.

The Taliban’s latest comments represent yet another twist in a chapter that has played out since their conquest of the country led tens of thousands to try to flee.

Like the United States, the Taliban do not want to see a repeat of the desperate scenes that played out in Kabul, where thousands of Afghans rushed to the airport and many were evacuated without proper documentation.

The Taliban are also working with international partners to try and restart full operations at the airport in Kabul.

Turkey’s foreign minister, Mevlüt Çavuşoğlu, said that his nation was working with Qatar and other countries to help fix damage at the Kabul facility and establish security protocols.

“The Kabul airport can be opened to international flights again, the damage can be repaired, the runway can be renewed, the terminal, also renewed,” Mr. Çavuşoğlu said, noting that 19 Turks were working to repair damage.

While the security outside the airport can be maintained by the Taliban, he said, the airport’s security “should be maintained by a security company that is trusted” by the international community.

“There are companies that do this business, if military presence is unwanted,” he said. Without such security, he said, it was unlikely that commercial flights would be able to resume.

“Even if planes want to fly, the insurance companies will not allow,” he said in an interview with a private Turkish broadcaster.

At the news conference in Doha, Qatar’s foreign minister, Mohammed bin Abdulrahman bin Jassim Al Thani, said his government had sent a technical team to evaluate the Kabul airport and expressed optimism that “we are about to get everything operational very soon,” including for commercial flights.

The airport may currently be capable of receiving commercial flights for “a limited time in the day,” Mr. Al Thani said, adding that it requires an equipment upgrade to operate as a standard international airport.

He added that his government was also negotiating with the Taliban to ensure security there.

For now, Mr. Al Thani said, Qatar is sending charter flights to Kabul bearing humanitarian aid on a near-daily basis.