Hodgson was appointed at the start of the season despite Dalglish telling Christian Purslow and Martin Broughton he wanted the job of replacing Rafa Benitez.
As Liverpool struggled for results during Hodgson's unpopular reign, supporters openly chanted for the return of club ambassador Dalglish to the dugout and got their wish in January.
Ahead of Saturday's game between West Brom and Liverpool, Hodgson, now manager at The Hawthorns, admits Dalglish's presence was an issue.
"It's difficult to compete with icons," Hodgson said. "I came to the right club, [but] perhaps I didn't come at the right time because Kenny did make it clear at that time that he wanted the job.
"But the people who were making the decision at that time decided to go for me. And of course, as a result, that left Kenny in a difficult position because he was the one who wanted the job so, when things didn't go well, having him in the background wasn't easy.
"I took the job in good faith. I knew I was taking a risk because there was a change of ownership in the offing and I knew that in order to win the fans over we would have to have a flying start.
"When you don't get that flying start and there is a change of ownership, I'm afraid you are at risk as a manager, especially when there is a man of Kenny Dalglish's iconic stature waiting in the wings and prepared to take over.
"But I have no regrets about going to the club. It was an honour and a privilege to work there. I would have liked to work longer, I make no secrets about that."
Hodgson also believes that Dalglish should be appointed Liverpool manager on a permanent deal, following the Scot's success in turning the club around and improving the standard of football at Anfield.
"Yes, I do, because the fans want him to have the job," he said. "He's got the backing of the fans and the fans are very important at Liverpool. If they don't give him the job now, it's going to be a very difficult job for the next man who gets it."