The statement, which was emailed to journalists, said the talks had taken place in Qatar, the New York Times reported.
"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."
U.S. diplomats confirmed that sensitive discussions had taken place, CNN reported.
Insurgents have held Bergdahl in Pakistan since 2009.
The statement said that the Taliban had provided the Americans a video of Bergdahl to prove he was still alive and in its custody.
U.S. officials told CNN that they are currently attempting to secure Bergdahl's release because of his poor health.
Authorities are also concerned that a full troop withdrawal from Afghanistan will make a rescue attempt difficult.
Other obstacles remain, as Congress would have to be notified of the five Taliban prisoners' release in advance.
Lawmakers have previously spoken out against it, CNN said.