An American man held hostage by the Afghan Taliban for more than a year is gravely ill and in urgent need of care, the militant group has warned as it urged the US to accept its demands.

Kevin King, 61, one of two professors at the American University of Afghanistan abducted at gunpoint in Kabul last year, is suffering from “a dangerous heart disease and kidney problem”, Taliban spokesman Zabiullah Mujahed said in a statement on Monday.

“His feet have begun swelling, he frequently loses consciousness and his health is worsening rapidly,” Mujahed said.

The Taliban did not provide evidence of King’s condition.

The last sign of life from the professor and his colleague, Timothy Weeks, came in January when the Taliban released a video of the two looking dishevelled and visibly distraught, pleading with the president-elect Donald Trump to negotiate for their release.

Western officials believe the two men are captives of the Taliban-affiliated Haqqani group, which until recently also held a Canadian-American couple for five years. Joshua Boyle and Caitlan Coleman were rescued by Pakistani security forces on 11 October, along with three children born in captivity.

The Taliban have not made public their demands for releasing King and Weeks, which the spokesman told the Guardian said had been shared with the US. However, in the video, the two men asked for a prisoner exchange.

Such a demand would probably involve Anas Haqqani, the brother of the Haqqani chief Sirajuddin Haqqani, who has been in Afghan government custody for three years, apprehended with a senior leader, Hafiz Rashid.

While it came into being before the Taliban, the Haqqani network has become an integrated part of the Taliban, with Sirajuddin Haqqani recently assuming the role of deputy leader overseeing armed combat.

Suspected of some of the most brutal and indiscriminate attacks in Afghanistan, the Haqqani group operates mostly in the eastern part of the country.

In a press statement, the American University of Afghanistan urged the Taliban to release their two teachers immediately.

“They are innocent victims of a criminal abduction. They came to Afghanistan to teach Afghan youth and contribute to building a peaceful Afghanistan. They have done no harm to anyone,” the university said.

Meanwhile, a Taliban commander in eastern Afghanistan told the Guardian the group considered teachers at the American University dangerous as they “change the minds of society”.

US forces have conducted at least one unsuccessful attempt at rescuing King and Weeks. Days after their abduction, a team of Navy Seals descended on a militant hideout in the mountains of eastern Afghanistan, missing the two men by a few hours, according to reports.

The Taliban said they had tried to treat King’s illness, “but since we are facing war conditions and do not have access to health facilities, we are unable to deliver complete treatment”.

“If [the US] insists on delaying this matter and the illness of Kevin King becomes incurable, or he loses his life, the Islamic emirate will not be held responsible,” the statement said.