The Daily Mail understands that Liverpool expect to announce him as their new boss next week.
It is believed initial contact has been made with Fulham and that chairman Mohamed Al Fayed has reluctantly accepted last season's manager of the year will be installed as Rafa Benitez's successor.
Hodgson, whose exploits with Fulham have put him in the frame to take over as England manager after Fabio Capello, is at the World Cup but has been made aware that Liverpool are primed to appoint him, ahead of Kenny Dalglish and Manuel Pellegrini.
Liverpool's board carefully considered the claims of both Dalglish, who had asked to be considered after acting as adviser to managing director Christian Purslow, and Pellegrini, who led Real Madrid to a record points tally in La Liga but lost his job to Jose Mourinho after finishing second to Barcelona.
But they concluded Hodgson was best suited to reviving their fortunes.