In a terse Pashto language statement emailed to the Associated Press on Sunday, Zabihullah Mujahed blamed the "current complex political situation in the country" for the suspension.
Bergdahl, of Hailey, Idaho, was last seen in a video released in December, footage seen as "proof of life" demanded by the United States. Bergdahl is believed to be held in the border regions between Afghanistan and Pakistan.
The video of Bergdahl was part of the negotiations which were to lead to the eventual transfer of five senior Taliban leaders held since 2002 in Guantanamo Bay.
"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," according to the statement.
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"The process will remain suspended without the exchange of the prisoners until our decision to resume."
Afghanistan is in the middle of a presidential election campaign. President Hamid Karzai cannot seek another term in office under the Afghan constitution which allows only two terms as president.
The election is scheduled for April 5. The US State Department has refused to acknowledge the negotiations, but a U.S. Official familiar with the negotiations confirmed to The Associated Press that indirect talks were underway. He requested anonymity because he was not authorised to speak to the media.