The 31-year-old is out-of-contract in the summer but Spurs are keen to take the left-back during this January transfer window.
"The door for Tottenham is now so wide open that you can talk about it," said Hertha general manager Dieter Hoeness to Sport1. "The transfer fee must be high enough so we have enough to commit to a substitute for Gilberto.
"The talks continue. It is and remains the decision of Hertha whether Gilberto moves or not."