South Africa-born Berg suffered the injury while playing against Nottinghamshire in Barbados.
The 35-year-old will be assessed in the next two weeks to decide whether he will need an operation.
"If there is an operation then you're looking at probably mid-season," said Benkenstein.
"I'm hoping it's only six weeks if he doesn't have to have an operation," he told BBC Radio Solent. "I never like operations and there's a chance he can get better without the operation."
Hampshire coach Dale Benkenstein says all-rounder Gareth Berg could be miss up to half of the upcoming season should he require knee surgery.