Apart from the Burnley game, he is also set to miss matches against Stoke City, Newcastle United, West Ham United, Arsenal and Chelsea in the home stretch of a difficult season for 18th-placed Southampton.
"Maya will be out for a minimum of four to five weeks," Pellegrino told a news conference on Thursday.
"Last week I said that everybody is OK and then in the last training session three or four players out," added Pellegrino, who also lost midfielders Steven Davis and Sofiane Boufal ahead of the West Brom game due to injuries.
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"Football has this unpredictable moment and you cannot control this but hopefully in the next few weeks everybody will be alright.
"Maybe Sofiane will be involved in the team (against Burnley), but Long and Davis will have to wait another week," the Argentine added.
Southampton have one victory in their last 14 league games and come into Saturday's clash on the back of a 2-0 defeat by Liverpool. Seventh-placed Burnley are also in the middle of a poor run with no wins in 10 league games.
(Reporting by Shrivathsa Sridhar in Bengaluru; Editing by Toby Davis)
