The 34-year-old has one year remaining at the Riverside but has turned down the initial offer of an extension to his contract.
Schwarzer, who is currently on duty with Australia at the AFC Asian Cup, told PA Sport: "I have had a great 10 years there. I am happy there and have been happy there all the time and there is no reason to go looking elsewhere. But by the same token you have to look after your future and sometimes people move on and sometimes the club moves on."
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