The 31-year-old remains an integral part of the Blancos team under Carlo Ancelotti but has now entered the final two years of his deal with the capital giants.
Having failed to sign a replacement for Georginio Wijnaldum in the summer, Liverpool are being tipped to bolster their midfield ranks next year, with Napoli's Fabian Ruiz thought to be on the Reds' radar.
The report adds Klopp and Kroos have already had several positive conversations, and the Reds plan to make an 'irrefutable offer' for the World Cup winner to make the move to Anfield.
Kroos has registered two goals and two assists from 13 games in all competitions this season and has won four Champions Leagues, two La Liga titles and four Club World Cups during his seven years at the Santiago Bernabeu. body check tags ::