Wickham, 17, suffered the problem during training on Monday when his ankle gave way while taking a shot.
Wickham first came to prominence last season when he scored twice in the League Cup win over Shrewsbury Town last season for his first senior goals.
His first Championship goal was a stoppage-time winner in the 1-0 win over Scunthorpe in March.
He had been linked with a move to the Premier League this summer, but the injury appears to have ended any chance of that.
Midfielder David Norris has a back injury meaning he will be left out of Saturday's friendly against West Ham United but there is no suggestion he will miss the start of the season.
Bristol City have had a bid for the Sheffield Wednesday goalkeeper Lewis Grant turned down, leaving Burnley the only club able to hold talks with the player.
He was expected to have a medical at Turf Moor on Tuesday, if he completes the move it would reunite him with Brian Laws - who was managed him at the Owls.