The attacking midfielder has made only seven Bundesliga appearances this season, three of which have come from the subs bench.
The 23-year-old recently revealed he had already been in contact with Klopp.
He told German television station ZDF: “We are texting now and again.
“He was my coach at Borussia Dortmund, but there is nothing special.”
Gotze has a year remaining on his current Bayern contract.