Durham are understood to be in advanced negotiations with the player, but the management is not willing to reveal the deal, according to a BBC report.
Head coach Jon Lewis believes that the former Sri Lankan skipper would be a tremendous addition to the Durham side.
"I can't think of a better player in the world who is not involved in the Indian Premier League," Lewis told the Northern Echo newspaper.
Sangakkara, 36, retired from T20 cricket on a high after the islanders won the World Cup in Bangladesh recently.
Durham have already signed Australian all-rounder John Hastings as an overseas player for the 2014 season.
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