Cole surprised many by signing for the Anfield outfit on Monday, having turned down offers from both Arsenal and Spurs.
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And Carragher, 32, revealed that he is delighted that his former England colleague has chosen Liverpool.
"It's great news and it is a message that Liverpool mean business," he told the club's official website.
"We have signed a top international player who has been really important for Chelsea.
"A lot of clubs wanted to sign him but it is a real coup that he has chosen to come here and you have got to give the manager and the board a pat on the back for convincing him to come here.
"Every supporter that I have spoken to is absolutely delighted and all the players are the same - it's given everyone a boost, just at the right time.
Carragher hopes that Cole’s versatility can be instrumental in the long-term progression of Roy Hodgson’s side.
"Joe is someone I have known for a long time from playing with England. He is a great lad to have around the squad, he mixes well and is a lively character - he's also an outstanding footballer.
"In the modern game, you need forwards that can function in two or three different positions and Joe will do that, no problem at all. He can play on the right, through the middle or on the left."
By Ian Woodcock
