The Hammers have announced that the duo, and George McCartney, are free to leave Upton Park while negotiations with several other players continue
Cole started his career at Upton Park before leaving for London rivals Chelsea in 2003 where he won three Premier League titles and three FA Cups.
An unsuccessful spell at Liverpool followed and the former England international also had a loan stint at French club Lille.
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The 32-year-old rejoined West Ham on an 18-month deal in January 2013, but the midfielder has failed to become a regular starter under Sam Allardyce's stewardship.
West Ham also confirmed on Friday that full-back George McCartney and midfielder Collison will leave the club.
Collison, a Wales international, spent part of this season on loan at Championship club Wigan Athletic.
Goalkeeper Jaaskelainen and versatile left-sided player Taylor are in discussions about extending their deal, while the club have activated a clause to extend the deal of defender Dan Potts.
A statement on West Ham's official website read: "The Hammers can confirm that Joe Cole, George McCartney and Jack Collison will be given free transfers when their contracts expire on 30 June.
"West Ham United would like to thank Joe, George and Jack for their considerable efforts and contribution to the success of the club in recent seasons and wish them well in their future careers.
"Discussions with goalkeeper Jussi Jaaskelainen and midfielder Matt Taylor are ongoing over new contracts, while the club has exercised an option to extend the contract of defender Dan Potts."