The striker joined West Ham from London rivals Chelsea in the summer of 2006 and played for them until the end of the 2014-15 season.
The Englishman scored 68 goals in 293 appearances for West Ham, including one in the 2011-12 Championship playoff final against Blackpool at London’s Wembley Stadium.
Allardyce worked with Cole at West Ham for four years from 2011 to 2015, and he has claimed that the 31-year-old did not get the recognition he should have at the Boleyn Ground.
“That season (2011-12) was special for all concerned — and to end it with that success at Wembley made it unforgettable,” the former West Ham manager told The Scottish Sun.
“I’ll always remember that team with great fondness and appreciation. Carlton played a big part for me and the club that season.
“He maybe didn’t get the credit he deserved. He scored important goals all through that campaign and was totally committed for the team.
“But before a ball had even been kicked he’d agreed a massive wage cut following relegation. For me that said a lot about the boy. He wanted to stay and do his bit for West Ham United.”
Cole is training with Scottish Premiership champions Celtic with a view to signing a contract.
The 31-year-old is not a prolific striker, and many will suggest that he is no longer good enough to play in the Premier League.
Moving to Celtic does not mean that the former Chelsea man will get to play regular football, but given his experience, he is certainly going to add something to Ronny Deila’s squad.