Cambiasso, 34, joined Leicester last August on a short-term contract that has now expired.
He helped them avoid relegation back to the second tier in his only campaign at the Midlands club.
"I would like to announce that I have decided not to renew my contract with Leicester City," he said in a statement 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."
Cambiasso was the club's player of the year last season after making 33 appearances and scoring five goals.
The midfielder joined Leicester after spending 10 years at Inter Milan where he won five Serie A titles and the Champions League.
He began his playing career at Independiente before spells at River Plate and Real Madrid.
He made more than 50 appearances for Argentina in an 11-year international career, and scored one of the goals of the tournament at the 2006 World Cup, finishing off a 26-pass move in an 6-0 rout of Serbia.
(Reporting by Toby Davis; editing by Ken Ferris)
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