Youngs was injured during Leicester's 45-0 defeat by Bath in the English Premiership at the weekend.
England face New Zealand, South Africa, Samoa and Australia in November.
"He had surgery on Monday and it will probably be three months," Cockerill told BBC Radio Leicester.
Youngs has 17 England caps and a Lions tour under his belt, having made his international debut two years ago against Fiji.
(Reporting by Martyn Herman; editing by Justin Palmer)
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