Pellegrino is currently in charge of Malaga, where he has impressed after good spells in charge at Real Madrid and Villarreal.
He could face competition from Jose Mourinho, according to the Daily Mail, who report that Chelsea are looking at their various options with the season drawing towards a close.
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The reputation of the Premier League means it is the most popular and one of the most watched leagues in the world.
Five English clubs will compete in the last sixteen of the Champions League for the first time ever as Liverpool, Manchester United, Manchester City, Chelsea and Tottenham face some difficult away matches.