Cole - surplus to requirements at Liverpool under Brendan Rodgers - trained with the Hammers on Thursday and is hoping to be available to face Manchester United in the FA Cup on Saturday.
The former England playmaker failed to make his mark at Anfield after joining on a free from Chelsea in 2010.
QPR boss Harry Redknapp, who gave Cole his competitive debut in a West Ham shirt in 1998, confirmed he had tried in vain to lure the 31-year-old to Loftus Road.
"I tried to get him to come to QPR but he's going home to West Ham," said Redknapp.
Cole left Upton Park for Chelsea in 2003, before joining Liverpool after failing to agree a new contract at Stamford Bridge.
Redknapp added: "I lost Joe who would have been my first signing. He probably looked here and thought 'Harry I love you but you're not going to stay up'.
"I like Joe and he would have been good for me. He's West Ham through and through and they love him there. I thought he was coming for sure but West Ham came in."