The Pole was captured by television cameras insulting the German champions' bench as he made his way from the field after being sent off for fouling Arjen Robben.
The goalkeeper was dismissed shortly before halftime in Wednesday's Champions League last 16 first leg tie at the Emirates Stadium, when he was adjudged to have denied Arjen Robben a clear goal-scoring opportunity.
Szczesny will definitely miss the return leg at the Allianz Arena because of the red card, though he could face a lengthier ban if Uefa decide to review the incident.
Arsenal manager Arsene Wenger was visibly frustrated with the decision, while Szczesny expressed his thoughts before walking down the tunnel.
Wenger confirmed the Poland international will be sanctioned by the club.
"It was frustration but we are professionals, me included, and we have always to master our reactions," he said. "Wojciech is part of that as well. We'll deal with that internally. We don't agree with that."
Szczesny is available for Arsenal's fixture with Sunderland at the Emirates Stadium on Saturday.