The Daily Express says Crouch made his first Premier League start of the season for Liverpool at Fratton Park on Saturday and, although he failed to score before being substituted, he still did enough to convince Redknapp he would be a valuable asset for Portsmouth.
But Liverpool are not keen on the prospect of Crouch playing for another Premier League club and may try to invite bids from abroad.
Juventus are reported to be keen on him and sources close to the player claim that he fancies the challenge of testing himself in Italy or Spain. But Redknapp said: "I'm a big fan of Crouchie. He's effective and difficult to pin down."