How come Harry Potter never looked up to Remus as a father figure?

I was just rereading Prisoner of Azkaban, and it just came to me that through losing Sirius with him going on the run, wouldn't it make sense for Harry to try to go to Remus and see him as a father figure seeing how close he was to his father? I mean, I'm sure Lilly and James wouldn't have minded making Remus his godfather if they had to. Oh well, I know its fiction, but still, nags my mind a bit.