October 22, 2015, 2:10 pmUpdated: October 22, 2015, 2:12 pm
The 32-year-old has signed a contract until the end of next season following a trial spell.
Cole was released by the Hammers in the summer after netting three goals in 26 appearances last season.
Cole has scored 85 goals in 384 games in England, the bulk of which came duringalmost nine years at Upton Park. He also won seven England caps.
The striker comes in as competition for Leigh Griffiths and Nadir Ciftci.
With Anthony Stokes completely out of favour, Celtic have been short of numbers up front since Stefan Scepovic returned to Spain just before the transfer deadline to sign for Getafe on a season-long loan.