Hudson-Odoi has been subject to multiple bids from the Bundesliga champions and is intent on escalating the matter to convince the English side to let him leave.
Bayern are eager to negotiate with Chelsea, and are willing to increase their offer, though they have yet to make that improved counter.
The German club previously tabled an offer of £35 million ($45.3m) and have promised to make the youngster a key part of Niko Kovac's rebuilding effort following the 2018-19 season. Bayern have even gone so far as to offer the No.10 shirt, currently worn by the departing Arjen Robben, to Hudson-Odoi to show his potential role.
Chelsea manager Maurizio Sarri had previous argued Hudson-Odoi would be best served by continuing his development with the Premier League club, which the teenager has been a part of since the age of eight.
Hudson-Odoi has been angling for the move in hopes of securing playing time, which has been difficult to come by at Stamford Bridge. The 18-year-old winger has only made three substitute appearances in the Premier League this season, with his only starts this season coming in cup competitions.
He has made two starts and a total of four appearances in the Europa League, while adding a single start in each of the Carabao and FA Cups.
Chelsea had been growing increasingly hopeful they had managed to ward off Bayern's interest with talk of increased playing time along with improved wages of £70,000-a-week in an attempt to commit him to a new deal beyond his current one, which expires in 2020.
In addition, the Blues supporters have aimed to help convince him to remain, chanting "Hudson-Odoi, we want you to stay" in the recent victory over Newcastle.
Instead, concerns over his role look to have him angling to follow in the footsteps of Borussia Dortmund's Jadon Sancho, who departed Manchester City and has emerged as one of the Bundesliga's top players this season with six goals and nine assists in the league.
And with five days remaining in the transfer window, it appears Hudson-Odoi is intent on seeing if he can be the latest in a trend of English youngsters making a name for themselves on the continent.