The highly-rated teenager has struggled to get game time at Chelsea this season and is said to be desperate for first-team action.
Hudson-Odoi, 18, has made just five appearances this campaign – with just one of those coming in the Premier League.
That was a substititue appearance against Watford on Boxing Day, where the winger came on as a sub only to be taken off again because of a hamstring issue.
The Daily Mail report that Bayern will start the bidding when the January transfer window opens, with Borussia Dortmund among several Bundesliga clubs interested in the young star.
Hudson-Odoi is one of the best talents to develop from the Chelsea academy in recent years and the Blues would reportedly insert a buy-back clause should any deal be done.
Bundesliga clubs have found great success with young English talent in the last couple of years, with Jadon Sancho one of the stars for Dortmund this term.
He has seven goals and nine assists so far this season and won a senior England call-up in November.
Arsenal loanee Reiss Nelson has six goals for Hoffenheim and Ademola Lookman impressed at Red Bull Leipzig last season.
Hudson-Odoi, who kept Sancho out of the England Under-17 side when the Young Lions won the U17 World Cup last year, has played only ten games in two seasons for the Blues.
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