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The Scottish Under-21 international limped off at the end of the first half against Preston North End on Sunday, having given Millwall an early lead, after picking up an ankle injury,
The 20-year-old, who is being watched by a number of clubs including Glasgow giants Celtic, may now miss The Lions last away game of the season.
"I'm gutted I've got another injury," Sweeney told the club's official website.
"It's the third or fourth I've had this season, which has been really disappointing, but I think it's just a sprain and I really do hope to be fit."