The German said: “He's made a brilliant start at Ibrox — then he lost against Celtic. Wow, what a surprise!
“Rangers are getting closer and that's the only way to get them one day.
“That's Steven's job, but so far it's more than good."
Klopp is in regular contact with his former skipper as Gerrard has three Reds in his Ibrox squad — Ovie Ejaria and Ryan Kent have joined on loan, while Jon Flanagan was signed permanently after leaving Anfield.
But Klopp stressed that, unlike other managers new on the scene, Gerrard doesn't have to ask him for advice — because he already has the knowledge.
“Rangers were brilliant in Europe — Gerrard is putting a good team together," he said.
“When I started out as a manager I didn't ask for advice because I didn't know who to ask. Steven doesn't ask because he doesn't have to.
“We are in contact, but not for advice."