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The German champions have already had three offers knocked back for the teenage forward this window, the most recent believed to be in the region of £30m.
Chelsea boss Maurizio Sarri revealed on Monday that he does not want to lose Hudson-Odoi, despite only using the youngster twice from the bench in the Premier League all season, but privately the Blues are said to be prepared to do business if his valuation is met.
According to The Sun, Bayern's latest bid is expected to be enough to complete a deal, with the England Under-19s international now free to discuss personal terms with the Bavarian giants.
Hudson-Odoi, who has 18 months left to run on his Stamford Bridge contract, is seen as Arjen Robben's long-term successor at Bayern.
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