The duo previously worked together at Borussia Dortmund.
The German manager signed the 27-year-old winger for Borussia Dortmund back in 2011, before selling him to Wolfsburg two years later in 2013.
The Croatia international has since joined Inter Milan, signing a five-year deal for a reported £14.5 million (The Mail) just last summer, but speculation is rife that he could be leaving Italy already.
Perisic, who netted nine times in 42 Bundesliga appearances for Dortmund, has attracted reported interest from the Toffees (Goal.com) and such a deal may hinge on whether the Nerazzurri secure Champions League qualification or not.
But Mediaset Premium have suggested that Klopp has entered the race to sign his former winger, who could cost in excess of £23 million.
During their time at Signal Iduna Park, Klopp once labelled the Croat as "childish" for airing certain comments in the media, quoted by Goal.com at the time.