But he has "not closed the door" on LA Galaxy striker Edson Buddle who has returned home after training with Blues for the past week.
Carew, who is out of contract in the summer, has struggled to make an impression during the first half of the current campaign and McLeish said: "I have always liked John Carew as a player and I tried to take him to Rangers. But he is not someone I have thought about."
As regards Buddle, McLeish said: "He has gone back and we will keep the door open.
"We are not in the position to make any offers at the moment because we are looking to try and bring a more experienced player into the club.
"We've not closed the door. He did okay."
McLeish is aware of the need for Birmingham to strengthen during January.
He believes the current financial climate will mean players being available.
He said: "We are looking at one or two situations and I am working closely with the board in trying to add to the squad. It is difficult to bring in players you don't know because it does become a risky game in January.
"Prices are inflated as well but I do sense there will be players available because a lot of teams want to off-load. I don't know if there is massive debt in the game but I certainly feel a lot of clubs are feeling the pinch."