The Demons' 2021 premiership skipper will have weekend scans to determine the extent of his injury after being forced out of training on Friday.
"Max unfortunately had some hamstring awareness towards the end of this morning's training session," Melbourne's general manager of AFL performance Alan Richardson said in a statement.
"He will be closely monitored and assessed over the weekend. We're hopeful we will have some further clarity early next week."
Melbourne's premiership season begins on March 18 but any lengthy lay-off for Gawn, a six-time All Australian, would be partially offset by the recruitment of former Collingwood ruckman Brodie Grundy late last year.
Essendon's top pick at the 2022 national draft Elijah Tsatas has also sustained a pre-season injury and will have surgery on a torn meniscus in a knee.
The 18-year-old, who was the fifth pick overall at the draft, copped a knock to a knee in a tackling collision at training this week.
Tsatas faces about two months on the sidelines.
"Whilst this setback is disappointing we won't be rushing his return to play," the Bombers' general manager of performance Dan McPherson said in a statement.
Meanwhile, St Kilda have officially signed ex-Carlton utility Liam Stocker, who was delisted by the Blues at the end of last season.
Stocker had been training with the Saints, who have signed 22-year-old Stocker as a pre-season supplemental selection.
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