The Bundesliga giants most recently submitted an offer of £30m to sign the teenager, who is out of contract in the summer and looking for assurances over first-team minutes.
Hudson-Odoi is understood to initially have needed some convincing over a move to Germany but has been impressed by Bayern's commitment to trying to bring him to the Allianz Arena.
Chelsea are keen to tie down Hudson-Odoi to a new contract but, with his deal expiring in June 2020, all the leverage is with the 18-year-old winger as things stand and he needs to see material progress before reassessing his stance though he told Chelsea TV he has "just got to be patient."
Bayern are set to overhaul their ageing squad in the coming off-season and have already begun their efforts to secure top young talent as they phase out the likes of Franck Ribery, Arjen Robben and Rafinha.
Canadian wonderkid Alphonso Davies has already signed with the club, while Timo Werner, Hudson-Odoi and Benjamin Pavard are among the players being targeted for a summer move.
Pavard, a World Cup winner, has refused to sign a new contract with VfB Stuttgart and is expected to sign for Bayern on a Bosman. Stuttgart, however, have spent the last week trying to find a club who will spend a fee on the France full-back, with Inter Milan interested.
Yannick Ferreira-Carrasco has been offered to the club, with the former Atletico Madrid winger keen to return from China.
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