As reported by Goal, the Bundesliga giants have tabled a fourth bid worth £35 million ($45m) for the 18-year-old winger, while they have offered him the No.10 shirt in an attempt to convince him to move to Germany.
But the Blues will act should they find that Bayern made any illegal approach to Hudson-Odoi.
'Tapping up' in football describes the illegal practice of clubs bending the rules to sign a player, which includes contacting the player, family or their representatives without the permission of the club holding their registration.
Chelsea have yet to make a decision over the latest offer for the 18-year-old but, with Hudson-Odoi stalling on renewing his contract at Stamford Bridge that expires in 2020, it would seem a switch to Germany is inevitable.
Should they decide to accept the bid, Hudson-Odoi would rank as the third biggest signing in Bayern's history, after Javi Martinez and Corentin Tolisso.
Indeed, Bayern sporting director Hasan Salihamidzic made clear that the club are interested in signing Hudson-Odoi, confirming to reporters on Wednesday that he is on their list of January targets.
"He's a very interesting player that we really want to sign," he said. "He has qualities that suit our game. He's a strong dribbler, fast, and carries a goal threat."
Chelsea are set for a busy transfer window beyond deciding the future of Hudson-Odoi, with Alvaro Morata, Davide Zappacosta and Cesc Fabregas all possibly leaving the club this winter.
The Blues are lining up replacements for Morata, Fabregas and Zappacosta - in the event that they all leave - with AC Milan forward Gonzalo Higuain, Napoli full-back Elseid Hysaj and Cagliari midfielder Nicolo Barella among those Maurizio Sarri's side are looking at.
Away from transfer matters, Chelsea return to Premier League action on Saturday when they face Newcastle at Stamford Bridge.