The Guardian says Liverpool are assessing their options to replace Rafa Benítez and have not made an official approach to Fulham for Roy Hodgson, who is due to depart to South Africa to commentate at the World Cup.
Dalglish remains interested in the job despite Liverpool's preference for a manager with more recent experience of Europe and the Premier League.
Hodgson does fulfil that criteria, and has support inside the Anfield boardroom, although Liverpool insist that, with players not due back for pre-season training until July, they are under no pressure to rush their search.