The German, 27, signed a one-year deal at the City Ground earlier this summer following his release by Hamburger SV.
With Forest currently one point clear at the top of the table, Tesche hopes to pen a new deal in the near future.
"I have only signed a one-year contract, but it is already in my head that I would like to stay longer," he told the Nottingham Post. "I would like a longer deal, but my first step is to acclimatise here, then we can see what happens beyond that.
"The reason it was one year was to give the club a chance to look at me and I can see if the move works for me. For both sides, it is a good option."
Forest return to Championship action on Sunday when they host local rivals Derby County.