Pellegrini was sacked by Real last month after crashing out of the Champions League and failing to eclipse Barcelona in the race for the La Liga title this season.
The Chilean’s representative denied reports that his client met with Dalglish at Heathrow Airport but Pellegrini was in England and did meet the Liverpool legend, who is in charge of naming a replacement for Benitez, according to The Guardian.
Talks with Pellegrini exposed that Liverpool are down to a short-list of potential managers, with Fulham boss Roy Hodgson remaining at the top of that list.
While Liverpool have not made a move for Hodgson as of yet, an official offer to the Fulham manager could be made as soon as next week.
Hodgson insisted that he is content at Craven Cottage and knows nothing of Liverpool’s interest in him.
"I'm happy at Fulham. It is all speculation and I can't enlighten you. I don't know any more than you do,” Hodgson said.
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