The 18-year-old England international suffered the injury during Monday’s 2-2 draw with Burnley at Stamford Bridge, and underwent an operation on Tuesday evening.
In a brief video message via his social media accounts, Hudson-Odoi said: “Hi guys, thank you for all the messages and support that you have been sending me, really appreciate it.
“I am doing better and feeling better, so thank you very much and (it’s) just the road to recovery now. Thank you.”
No time frame has yet been given for Hudson-Odoi’s rehabilitation.
The winger, though, will now miss Chelsea’s remaining fixtures as they bid to secure a top-four spot and also win the Europa League.
Hudson-Odoi would also have been expected to be involved for England’s Nations League finals this summer after making his debut last month.
A product of Chelsea’s academy system, Hudson-Odoi made 28 appearances and scored five goals in his breakthrough season.
Opperation was Successful, Thank God always and thank you all for your messages and support,the return begins now!!!❤️❤️🙏🏾 pic.twitter.com/14qw4rgRK2— Callum Hudson-Odoi (@Calteck10) 24 April 2019
The winger made his debut in the Community Shield against Manchester City in August and his emergence as a first-team player attracted the attention of England boss Gareth Southgate.
Hudson-Odoi was handed his first senior cap against the Czech Republic and then was named in the starting XI for the 5-0 win in Montenegro.
The teenager also attracted interest from German giants Bayern Munich during the Janauary transfer window, with a £35million offer said to have been on the table.
The United manager said he expected City to “snap at their heels” and foul his side in Wednesday’s crunch Premier League derby at Old Trafford.
Solskjaer is not the first United boss to have claimed City might strategically foul opponents high up the pitch to prevent counter-attacks, with his predecessor Jose Mourinho having mentioned it last season.
“When we win it we have got to be ready for their aggression because they will snap at your ankles and heels and kick you,” Solskjaer said at his pre-match press conference.
“They will not allow us easy counter-attacking. There will be fouls, no doubt about it.”
Guardiola reacted with surprise when this was mentioned at his own media briefing.
“Did he say that?” the Catalan asked. “I never prepare a game in 10 seasons as a manager thinking about these kind of things, never.
“I know exactly what I said to my players from day one to the last day. When a player wants to attack, we have to be honest.
“Of course there is contact, there are fouls, but when it happens (it is because) you arrive late and that is why there are referees, to make yellow cards or red cards or whatever they decide.
“I prepare to do our own game, that is what I want, knowing of course the opponent, but I never said I’m going to do that to punish them or cancel them (by) making fouls. Never.”
Guardiola was asked if he thought comments such as Solskjaer’s could influence the referee.
“That’s the reason why, of course,” he said.
Guardiola also claimed that with an “average of 65 or 70 per cent” possession City do not commit many fouls. According to Premier League statistics, United have committed 190 fouls in the opponents’ half this season compared to 170 for City. United have accumulated 64 yellow cards and four red, against 38 and one for City respectively.
On the subject of yellow cards, Solskjaer felt City’s fouls do not attract as much attention because they are often high up the pitch.
The Norwegian said: “You don’t get the yellow cards, do you? But that’s just because they commit so many players forward and you can clearly see that they’ve got them in that mould of trying to win the ball back, and they do make fouls.
“It’s up to us to play through that press, be ready enough, play one and two-touch, don’t give them time… It’s not my decision, it’s the referee’s responsibility to do that.”
Manchester City manager Pep Guardiola is expecting to face “the best United” in the Manchester derby on Wednesday after Ole Gunnar Solskjaer’s side lost 4-0 to Everton on Sunday.
With just four games left in the Premier League season, Guardiola believes City need to “win all four games to retain the title.”
City are two points behind leaders Liverpool in the table but have one game in hand over the Merseyside club.
Manchester rivals United are on the hunt for a top-four finish, having slipped to sixth after their recent defeat.