Adams impressed in his first hit-out for the Swans in last Thursday's pre-season loss to Brisbane, having made a shock move from the Magpies during the trade period.
But the 30-year-old hurt his left knee under the weight of a tackle from the Lions' Cam Rayner during the third quarter and was immediately put on ice.
Sydney officials had hoped Adams would be passed fit to play in the season-opener against Melbourne at the SCG on Thursday night.
But the extent of the damage was worse than feared.
"Taylor Adams had scans last night which confirmed medial ligament strain in his left knee which he suffered in the game against Brisbane," the club said in a statement on Saturday.
"He is expected to miss 3-4 weeks."
Adams spent 10 years with Collingwood and played in their 2018 grand final defeat before enduring further heartbreak when he missed out on last year's premiership with a hamstring injury.
The 206-game veteran then shocked the AFL world with a trade request, moving to Sydney in search of more midfield time.
Adams will miss games against Melbourne, Collingwood, Essendon and possibly Richmond, with his Swans debut potentially coming against West Coast next month.
Former Swans captain Luke Parker (centre) has undergone surgery on his broken arm. (Dan Himbrechts/AAP PHOTOS)
He will have to wait until at least round 22 before facing the Magpies.
Sydney veteran Luke Parker is also expected to miss about a month of football after undergoing surgery on his broken arm.
Former captain Parker was injured during the last quarter of the Swans' practice match against Greater Western Sydney on February 22.
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