The Premier League clubs are looking to sign the striker in a cut-price January deal, according to The People.
The 20-year-old is regarded as one of Spain’s hottest young talents and is also being linked with a move to Juventus.
Isco has four years left on his current contract at Malaga, but has a £16million buyout clause in his deal.
Malaga boss Manuel Pellegrini has admitted he expects Isco could leave the club.
He said: “He is a player who has a great future. Nobody knows what his limit is.”
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Liverpool have unveiled their new home kit for next season, sticking with the hugely popular retro styling of the current shirt.
The reputation of the Premier League means it is the most popular and one of the most watched leagues in the world.