The Reds owners, Fenway Sports Group, are reportedly running out of patience with Hodgson after the abject display in the home defeat to Wolves on Wednesday which leaves them just three points above the drop zone.
Kenny Dalglish is thought to be the first choice to take over the reins if Hodgson's short tenure at Anfield comes to an end after a stuttering first half of the season.
Hodgson knows he's under pressure but says he has the full backing of the players, although he admits the same cannot be said of the fans. The 63-year-old said: "I have been lucky. The support I have had has been from the players and within the club.
"I haven't had a lot of support from the fans ever since I came here.
"The fans have not been happy with what they have seen in the whole of 2010."