Chelsea have yet to accept the offer and have been hoping to renew their 18-year-old's contract, but they have not been able to come to an agreement with Hudson-Odoi, who had grown unhappy with a lack of first-team chances ahead of the transfer window.
Hudson-Odoi is understood to still be in contention for the first leg of the Carabao Cup semi-final against Tottenham on Tuesday night. He is likely to be involved in the game at Wembley as Ruben Loftus-Cheek faces a spell on the sidelines, and with Pedro and Willian unlikely to be able to complete 90 minutes.
Chelsea will still decide the destiny of Hudson-Odoi, but with only 18 months left on his current contract, the Premier League club may decide that the offer from the German champions is too good to pass up.
Juventus, Real Madrid and Borussia Dortmund had earlier registered an interest in Hudson-Odoi, but it has become clear that Bayern are most committed to their approach and they are willing to offer him the number 10 shirt that he prefers to wear.
Chelsea have tried to ward off interest in Hudson-Odoi in successive pre-match press conferences, as Maurizio Sarri and Gianfranco Zola from the coaching team lauded their youngster's abilities . Sarri said on Monday that it would not be a good move for Hudson-Odoi to go to Bayern.
"I don't think so, because he's an English player, he's very young," Sarri told reporters at Cobham Training Centre . "He has the future and I think he has a very great future here, with the national team and with Chelsea. I think to stay here is better for him.
"I know, for me, he is a very important player. He's young, but he's improving. He played really well in the last match."
"[Hudson-Odoi] is from the same year as [Borussia Dortmund's Jadon] Sancho, which was very successful," he explained to reporters on Saturday.
"It is completely normal that we are aware of and are following the player. For if and when something will happen you will have to ask Brazzo (Bayern's sporting director Hasan Salihamidzic)."