The Germany international looks set to leave Schalke in the summer when his contract expires.
Goretzka has no intention of signing a new deal and his representatives are actively looking for Premier League suitors.
Manchester United, Manchester City, Chelsea and Tottenham are all showing interest in the 22-year-old but the Gambling Times claim Liverpool and Arsenal are leading the chase.
They state Jurgen Klopp is in regular contact with the 22-year-old and he is working to convince him to come to Anfield.
Meanwhile, Arsene Wenger is on the lookout for fresh attacking talent with Mesut Ozil and Alexis Sanchez looking likely to leave at the end of the contract.