The Poland international goalkeeper suffered the problem during his side's 3-1 loss at Stamford Bridge on Sunday, and adds to coach Pochettino's growing injury list that includes Morgan Schneiderlin, Guly do Prado and Kelvin Davis.
With keepers Boruc and Davis both on the sidelines, Pochettino is set to hand 21-year-old Paulo Gazzaniga his first Premier League start since last December against Aston Villa on Wednesday.
Southampton go into the clash with Paul Lambert's men on the back of successive league defeats at Arsenal and Chelsea, but their Argentine boss insisted he is looking forward rather than back.
"Artur has an injury in his hand, it's difficult to analyse how many weeks Artur will be out for but we do know he will be out for some weeks," Pochettino said. "We have a few players who won’t be fit tomorrow.
"We don't have time to dwell on our two consecutive defeats because we're only thinking about tomorrow's game.
"We've lost on two very difficult grounds at Arsenal and Chelsea. Now we'll try to impose our values and get the win tomorrow.
"I'm expecting a physical game. Aston Villa are a strong side and very good on the counter-attack, especially away from home.
"We're fully aware that they are a very dangerous side, so it'll be a very tough game tomorrow."
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