The Brazilian was withdrawn during the Blues’ 2-0 loss to Tottenham Hotspur on Sunday.
Chelsea play Leeds United on Saturday before welcoming Borussia Dortmund to Stamford Bridge on Tuesday.
Silva is having scans on his left knee and will be unavailable for both games.
It is a blow for the west London club amid their poor run of form.
The Blues have won just two of their last 15 games across all competitions.
Manager Graham Potter is under growing pressure despite the club owners backing him.
Chelsea are 10th in the Premier League after 24 games, with 14 points between them and the top four.
They lost 1-0 in the first leg of their Champions League round of 16 tie against Dortmund.
Potter’s side face a huge task to turn things around in the second leg.
Silva’s unavailability could prove catastrophic given his huge experience and quality.
The 38-year-old has played 27 games across the Champions League and Premier League this term.
He was recently rewarded with a one-year extension for his brilliant performances in the club’s colours.
Chelsea omitted French centre-back Benoit Badiashile from their Champions League squad.
However, they boast quality squad depth in defence and would not have an issue with selection.
Wesley Fofana replaced Silva on Sunday and could play alongside Kalidou Koulibaly again for the next two games.
Potter may have to win both to retain his job, and he has to do it without his best centre-back.
- Aston Villa 1-0 Arsenal: Bailey goal extends Villains’ unbeaten run at Villa Park
- Pep Guardiola denies City are reliant on Rodri but the stats disagree
- Preview: Aston Villa vs Arsenal – prediction, team news, line-ups
- Man Utd boss Ten Hag scoops Premier League MOTM award
- This Week In Football: Last-minute UWNL drama, Santos relegated & more
- Morning Mix: Arsenal face acid test at Villa, Chelsea boss has no excuses
- Arsenal or Liverpool: Who is better placed to beat Man City to the title?
- Copa America 2024 draw: Argentina to face Chile, Brazil and USMNT discover their groups
- Four reasons PSV Eindhoven are overtaking Ajax on and off the pitch
- Matip out for the rest of the season – what do Liverpool do now?