Hudson-Odoi has played seven times this season and has 18 months left on his contract.
The offer meets Chelsea's valuation of Hudson-Odoi, Sky Sports News understands, and does not include a buy-back clause.
The young winger featured for the whole game in Chelsea's Carabao Cup Semi Final First Leg loss to Tottenham last night, in a move that many read as Sarri trying to convince the England worldwide to stay.
'It's not easy getting in the first team at clubs of this size and I think it is, at the moment, all credit to Callum with so much going on around him at such a young age to be going out on the pitch and putting in the performances.
"He is a good player, but he is very young", he said.
Although Bayern are regarded as one of the heavyweights of European football, Alonso feels Chelsea can match them if they Hudson-Odoi is prepared to be patient and wait for his chance. "I can advise him but at the end of the day he decides his future". His sole Premier League appearance so far this season was as an injury replacement in a 2-1 win at Watford on December 26. You can see it from both sides. He can develop here but he can develop in Germany as well - and don't forget Bayern Munich are also a big club.
Chelsea star Callum Hudson-Odoi has warned the club they risk losing him on a free if they block his wish to join Bayern Munich.
'As someone from our academy I would obviously love him to stay and the dream for all of us at the academy is to see our boys playing in the first team, ' he said.