Hodgson was the most likely candidate to replace England manager Fabio Capello if the Three Lions had crashed out of the World Cup on Wednesday and the Italian was forced to resign.
The Cottagers' manager has been the Reds number one target to replace Rafael Benitez, who left Anfield to lead Inter Milan next season, and Liverpool made an official move for Hodgson last week, according to The Mirror.
Negotiations between Hodgson and Liverpool have stalled as the Fulham boss is currently working as a BBC pundit at the World Cup.
However, Hodgson is returning to England to complete his move to Liverpool, with club officials planning to announce the deal within the next two days.
Liverpool have talked to numerous candidates including Aston Villa boss Martin O’Neill, former Real Madrid manager Manuel Pellegrini and Alex McLeish of Birmingham City.
Reds legend Kenny Dalglish and managing director Christian Purslow were given the responsibility of finding a new manager and the two decided that Hodgson was the best choice for Liverpool.