Kabul, Afghanistan (CNN) -- The Taliban say they are suspending talks with the United States to exchange Army Sgt. Bowe Bergdahl -- the only American soldier held as a captive -- for five Taliban prisoners.
U.S. diplomats had been carrying out "extremely sensitive discussions" with Taliban intermediaries, a U.S. official told CNN on Tuesday.
The Taliban said the discussions were taking place in Qatar, where the group opened an office last year.
However, on Sunday, Taliban spokesman Zabiullah Mujahid sent a statement saying the group has called for the talks to be suspended.
"The leadership of the Islamic Emirate has decided to suspend the process for some time due to the current complex political situation in the country," the statement said. "The process will remain suspended without the exchange of the prisoners until our decision to resume."
Mujahid did not say what "complex political situation" the statement referred to.
Bergdahl has been held by insurgents in Pakistan since 2009.
Taliban leaders 'appreciate' Karzai
Another statement released by a Taliban spokesman Sunday said a high-ranking delegation recently met in an "atmosphere of peace and sincerity" with representatives of the Afghan High Peace Council in the United Arab Emirates.
The Taliban representatives "appreciate" Afghan President Hamid Karzai's release of 65 prisoners in February, the statement said. It credited the release "as a reply to our previous demands in this regard."
U.S officials strongly objected to the release of men who they said pose a threat to security forces and civilians. The group included an alleged Taliban explosives expert, a suspected Haqqani network commander and a specialist accused of building and placing improvised explosive devices.
"These individuals are dangerous," U.S. State Department spokeswoman Marie Harf said. "They pose threats to the safety and security of the Afghan people and the Afghan state."
Negotiating with terrorists
American officials are adamant that they have never given up on trying to get Bergdahl back and resist calling the current effort "renewed" discussions. But driving the current effort, according to the U.S. official, is Bergdahl's poor health and concern that a full troop withdrawal from Afghanistan at year's end would essentially close the door on any rescue attempt.
The United States has long declared that it won't negotiate with terrorists, which is what it considers the Taliban. However, "we have a history of being able to dance pretty effectively along the edges of the declaration," said retired Maj. Gen. James "Spider" Marks, a CNN military analyst.
One possible obstacle to securing Bergdahl's release is what may be asked in return. The Taliban have long demanded the release of five detainees at the U.S. military prison at Guantanamo Bay, Cuba, but another U.S. official said that releasing them would be difficult because Congress would have to be notified in advance, and lawmakers have previously resisted releasing them.
Van Hipp, who served as a deputy assistant Army secretary under former President George H.W. Bush, said that trading Guantanamo inmates for Bergdahl would put at risk "every American soldier deployed all over the world."
"We are sending the message to terrorist organizations all over the world that it's OK to capture an American soldier, that America will deal with you," said Hipp, now a defense consultant in Washington.
"I think we need to give the green light to the Special Ops and let them do their jobs quietly and methodically, as they have done so well in the past," he added.
Fits and starts
In May 2012, the U.S. government acknowledged that it had been engaged in talks with the Taliban to free Bergdahl, but those talks moved in fits and starts because of U.S. concerns that any Taliban prisoners swapped for the sergeant might be repatriated and allowed to rejoin the fight.
Bergdahl, of Wood River Valley, Idaho, appeared in diminished health in a video that the U.S. military obtained in January. His family also received a letter from him last year via the Red Cross.
The United States believes that one route to getting Bergdahl back may be through Qatari officials, who have been talking to the Taliban. But Bergdahl is believed to be held by operatives from the Haqqani network, an insurgent force affiliated with the Taliban and al Qaeda -- and it was not clear whether Haqqani operatives would abide by any agreement among the United States, Qatar and the Taliban.
CNN's Nic Robertson, Matt Smith and Barbara Starr contributed to this report