Kenny Dalglish will now manage the Merseyside club for the rest of the season, starting with the FA Cup third round tie against Manchester United this weekend.
The former Fulham boss arrived at Anfield in the summer after guiding the Cottagers to the Europa League final.
But Liverpool have struggled under Hodgson and, following a 3-1 defeat to Blackburn Rovers, sit a disappointing 12th in the Premier League table.
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"We are delighted that Kenny Dalglish has agreed to step in and manage the team for Sunday's FA Cup tie at Old Trafford and for the remainder of the season.
"Kenny was not just a legendary footballer, he was the third of our three most successful managers - three giants. We are extraordinarily fortunate and grateful that he has decided to step in during the middle of this season."
Hodgson, meanwhile, has expressed his disappointment at failing to rise to the challenge of managing the illustrious club.
"Being asked to manage Liverpool Football Club was a great privilege," he said. "Any manager would be honoured to manage a club with such an incredible history, such embedded tradition and such an amazing set of fans. Liverpool is one of the great clubs in world football.
"I have, however, found the last few months some of the most challenging of my career. I am very sad not to have been able to put my stamp on the squad, to be given the time to bring new players into the club in this transfer window and to have been able to be part of the rebuilding process at Liverpool.
"The club has some great, world-class players, with whom it has been a pleasure to work and I wish the entire squad well for the rest of the season.
"I thank those with whom I have built up a close working relationship at the club for their loyalty and support during very testing times, and finally of course to the Liverpool fans, your passion and dedication to the club will see Liverpool at the top of the game once more."