The 34-year-old, who played a key role in Sarries' European Rugby Champions Cup run last season, sustained a hip injury in the north Londoners' 46-14 home loss to Clermont on Monday.
The defeat, which saw Clermont gain revenge for Saracens' epic final victory at Murrayfield last May, was their sixth consecutive in all competitions and marked the first time the club had suffered such ignominy in 14 years.
Mark McCall's men must contend with a mounting injury list as they next head to France for the return leg on Sunday, with Wigglesworth joining centre Brad Barritt on the sidelines after the captain was concussed on Monday.
Maro Itoje, Billy Vunipola, Liam Williams and Schalk Brits will also be missing for a match crucial to Sarries' ambition of matching Toulon's achievement of winning a hat-trick of titles.
McCall admits that the current situation is as bad as he has known in his time at the club.
"It's obviously a big challenge," said the director of rugby. "We're not a club that over-reacts, but neither are we a club that sweeps things under the carpet either."