The former Argentina international played a pivotal role in helping keep the Foxes in the Premier League last season after joining on a one-year contract from Inter Milan.
New Leicester boss Claudio Ranieri revealed he had been in contact with the 34-year-old regarding an extension of his stay at the King Power Stadium but Cambiasso has decided his future lies elsewhere.
“I would like to announce that I have decided not to renew my contract with Leicester City,” Cambiasso said on his Facebook page.
“I have taken this decision only 24 hours after receiving the proposal from Leicester because I want to give the club as much time as possible to find an alternative.
“I would like to thanks all the fans who last season made me feel proud to be part of a club with an unforgettable spirit and energy.
“And a very special thanks to all my team mates, staff and every people of the club, with whom I have lived one of the most important years of my career, crowned with the remaining EPL status.
“Many thanks to everyone, keeping you all for ever in my heart Leicester City Football Club.”
Director of football Jon Rudkin had previously said Leicester were optimistic about getting a deal done but were aware that Cambiasso was likely to receive other offers.