The 32-year-old was given an 18-month deal in January of last year but he has not featured much during the 2013/14 campaign.
Cole scored in a 3-3 draw against West Brom in December but that was the last Premier League match he started.
However, he plans to continue playing and 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 and 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 is a graduate of the West Ham academy and is considered one of the most talented players of his generation.
He joined Chelsea in 2003 and won three Premier League titles during a spell in which he became an England regular, and he has represented his country 56 times at senior level.
Injuries led to him slipping down the pecking order at Stamford Bridge and a move to Liverpool in 2010 failed to re-ignite his career but he had a successful loan stint with Lille before re-joining the Hammers.