The Daily Mail says Klinsmann's Bayern were beaten 1-0 by minnows Hanover 96 on Saturday, leaving them floundering in ninth place with two wins, two draws and two defeats from their first six matches under the new manager.
Munich chairman Uli Hoeness left Klinsmann in no doubt that the pressure is on him.
"I'm annoyed now when I look at the table," Hoeness said. "We were much too careless."
If Bayern's Champions League campaign goes the same way as the Bundesliga, Klinsmann might soon find Spurs chairman Daniel Levy making contact.