The gestures are said to have been made at the end of Sunday's derby clash at Villa Park.
And the Football Association have confirmed they are aware of the incident and will study referee Martin Atkinson's match report and other evidence before deciding whether to take action.
Carr and the Blues players were left incensed by the award of a controversial late penalty to Villa which doomed the St Andrew's club to a 1-0 defeat.
Carr was at the centre of the protests and it is alleged he made a gesture towards Villa fans as he left the pitch.
A police spokesman told the Birmingham Mail: "West Midlands Police have received a complaint regarding an offensive gesture allegedly made by a player at the end of the Aston Villa-Birmingham City match on Sunday.
"The matter is currently being investigated by detectives and inquiries are ongoing."
A Football Association spokesman said: "We are aware of the incident and will be looking at the referee's report and other available evidence before we decide what to do."
A City spokesman said: "We are aware of the incident and we are looking at it internally."
If the FA decide to take action, 33-year-old Carr, who was persuaded last season to come out of retirement by Birmingham manager Alex McLeish, could face a suspension or a fine.
Carr insists he has no regrets over his actions during the controversial defeat.
Former Republic of Ireland international Carr admits he lost his head during the game, but he has no regrets over his behaviour.
"What was I doing by the spot?' said Carr. 'There was something on my shoe I had to get off! No, I was that frustrated," admitted Carr.
"Do I regret it? No. Not at all. That's life. You make decisions.
"It's hard not to react like we did. I'm absolutely gutted. It felt like we were robbed.
"I got booked for not moving away from the area but my head had gone, to be honest."
Carr added: "Roger Johnson got the ball. I explained to the ref that you have to see it, you have to be 100 per cent sure if you are going to give a penalty.
"I know sometimes you get them, sometimes you don't. But from his distance he could not see whether Roger had got contact with the ball."