Staunton's absence is of further worry to the Tigers with England fly-half Toby Flood already out injured and not expected to return to action until the middle of October.
"The initial scan showed a bit of swelling but there is a small tear in the Achilles," said head coach Richard Cockerill.
"He could have played with it but it's too risky and will have to be off his feet and rested up to 12 weeks."
The double injury leaves the Tigers short of options at outside-half ahead of their visit to Benetton Treviso on Saturday, October 9 for their opening pool game.
Billy Twelvetrees, the surprise hero of last season's Heineken Cup draw with the Ospreys, is expected to wear the No10 shirt in Italy, although England Youth star George Ford could also force his way into contention.
The same two players are expected to battle it out a week later when the Scarlets travel to Welford Road in Round 2 as the Tigers go in search of European success for the first time in eight years.