Liverpool have a deal in place with Bayern Munich for the midfielder and are determined not to lose a minute on deadline day.
Gravenberch also wants the move to proceed without an issue and he is, according to German daily Bild, is flying in.
The Dutch midfielder has flown in on a Bombardier Challenger 604 private jet and landed at 07:28 this morning.
Gravenberch boarded the plane in Munich shortly after 06:00 in the morning and then completed the short flight to Liverpool.
Liverpool will now set about conducting a thorough medical and if he comes through without an issue he will be able to sign a contract.
The Reds are paying Bayern Munich a fixed fee of €40m.
Gravenberch was highly rated when Bayern Munich snapped him up from Ajax, but failed to make an impact at the Bundesliga giants and now they are cashing in on him.