The winger twisted his knee during the final pre-season warm-up match against Doncaster on Saturday and limped off after 65 minutes.
New signings Gary Teal, Chris Sedgwick and Daniel Jones could all take up Johnson's wide midfield berth, although former Wolves man Jones is primarily a left-back.
The Owls have been one of the busiest sides in the Football League during the summer transfer window and Neil Mellor and Clinton Morrison could provide an eye-catching debut partnership up front.
Former Manchester City and England Under-21 goalkeeper Nicky Weaver became the latest arrival when penning a one-year deal on Thursday.
Weaver is expected start between the posts following Lee Grant's summer move to Burnley.