The 21-year-old Austria international limped off in their match against Wycombe Wanderers on Saturday after sustaining the knock in a crunching tackle.
Following a meeting with a club doctor on Sunday, it is thought that he will spend up to a fortnight on the sidelines as Villa continue their pre-season preparations.
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“Hopefully he’ll be okay in – I don’t know how long – in the next few weeks or so.”
The injury comes as a big blow to the Premier League side with Weimann standing out in their previous campaign, scoring seven goals in 30 appearances.
Despite bringing in a number of players in the close-season - including Nicklas Helenius, Leandro Bacuna, Jores Okore, Antonio Luna and Aleksander Tonev - a rising injury tally is becoming a concern for Lambert.
"We've got too many [injuries] at the minute," he added.
“There’s a few knocks we could do without and that’s a disappointing thing.
“There’s a few back at the training ground that we’ll hopefully get back in the next couple of weeks or so and we’ll see how they fare. Once we get them all back fit we’ll be pretty strong.”