No deal has been agreed between the clubs yet but the Netherlands international is thought to be willing to move to Anfield as he looks for first-team football.
Gravenberch joined Bayern from Ajax last summer but the 21-year-old has only started three matches in the Bundesliga since then.
Jurgen Klopp has already bought three midfielders, in Alexis Mac Allister, Dominik Szoboszlai and Wataru Endo, in this transfer window.
However, Liverpool have lost six other midfielders since the end of last season, in James Milner, Naby Keita, Alex Oxlade-Chamberlain, Arthur Melo, Jordan Henderson and Fabinho.
After submitting a British record £111m bid for Moises Caicedo, who instead chose to go to Chelsea, Klopp still has funds left in his budget.
Join our commenting forum
Join thought-provoking conversations, follow other Independent readers and see their replies