He was sacked after defeats by Wolves and Blackburn and a disappointing 25 points from 20 games.
Hodgson said: "I came to the right club, perhaps I didn't come at the right time because Kenny (Dalglish) wanted the job.
"So when things didn't go well, having him in the background wasn't easy and wouldn't have been easy for any coach. It wouldn't be strictly true to say I enjoyed my time at Liverpool but I respected my time there."
Asked if he thought Dalglish should get the job full-time, Hodgson replied: "Yes I do. I'm sure he will. The fans want him to have the job. He's got the backing of the fans and they are very important at Liverpool.
"If they don't give him the job now, then it's going to be a very difficult job for the next man who gets it."