The 26-year-old sustained the injury during last weekend’s opening day win over Nottingham Forest, where Fabian Schar and Callum Wilson were on the scoresheet.
Dan Burn started in place of Targett at left-back against his former club, while Sven Botman came in to partner Fabian Schar at centre-back.
It’s the first game Targett has missed since joining on loan in January before making his stay a permanent one for £15million from Aston Villa.
Elsewhere, Emil Krafth remains on the sidelines with a back problem, as does Jonjo Shelvey, who is expected to be out for three months after hamstring surgery.
Jamal Lewis and Federico Fernandez, after missing pre-season, returned in training this week but are short of match fitness.