The centre back limped off in the first half of Liverpool's 3-0 defeat to Watford on Sunday and his absence is likely to cause further headaches for manager Juergen Klopp and his struggling side.
"I will work hard on my recovery and will do everything to come back as soon as possible and even stronger than before," Skrtel said on Instagram.
The 31-year-old has been pivotal for Liverpool this season, appearing in all their Premier League games.
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His injury has come at a bad time for Klopp's side. They are in ninth place in the Premier League after taking one point from their last three league games and are already without centre back Dejan Lovren through injury.
France defender Mamadou Sakho returned to the heart of Liverpool's defence for the clash at Watford after his own lengthy injury break, but looked short of match sharpness.
A six-week absence would rule Skrtel out of league games against Leicester City, Sunderland, West Ham United, Arsenal, Manchester United and Norwich City.
He would also miss both legs of the League Cup semi-final against Stoke City and the FA Cup third round tie against Exeter City.
(Reporting by Toby Davis; editing by Stephen Wood)
