The 32-year-old Londoner returned to Upton Park on a free transfer in January 2013 after a disappointing spell at Liverpool.
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He told talkSPORT: "I'm a free agent now and I'm looking for a new club. My relationship with West Ham is great, but obviously it didn't work towards the end of this season. I didn't play enough.
"So I'm just going to take my time over the summer and find the right fit.
"At this stage of my career finances are not important for me, I want to enjoy my football, I want to compete and I just want to play more."
Cole, who started his career at West Ham, has been linked with a move to Championship side Brighton and Hove Albion.
He boasts 56 caps for England, but last represented his country in 2010.