United made an approach for Gravenberch earlier this summer but Liverpool stepped up their efforts in the closing stages of the window.
Jurgen Klopp was keen to secure a new midfielder after missing out on Moises Caicedo and Romeo Lavia, who both subbed Liverpool to join Chelsea.
‘I’m very happy. Very happy that the deal is done and I’m finally here,’ Gravenberch said after completing his move to Liverpool.
‘If you see it from the outside, it’s one of the biggest clubs in the world. Also the fans, the stadium, I think everything from the outside is top. So that’s why [it was the right club].’
Gravenberch had appeared just once for Bayern across their three games so far this season, coming off the bench for the last nine minutes against Augsburg.
He made a total of 34 appearances for the Bundesliga side, having only joined Bayern last summer from Ajax.
Gravenberch becomes Liverpool’s fourth signing of the summer following the arrivals of Dominik Szoboszlai, Alexis Mac Allister and Wataru Endo.
It has been a summer of change for Liverpool’s midfield after the club offloaded Fabinho, Jordan Henderson, Naby Keita, James Milner and Alex Oxlade-Chamberlain.
The Reds are unbeaten from their first three Premier League games of the new campaign, drawing at Chelsea before wins over Bournemouth and Newcastle.
Klopp’s side are back in action on Sunday as they host Aston Villa, looking to return to the Champions League next term after finishing fifth last season and ending up in the current Europa League.