Drogba appeared to fling a coin into the crowd after scoring in Chelsea's shock Carling Cup defeat at Stamford Bridge, claiming it had been thrown at him by Burnley fans.
A police spokesperson said: "We are investigating the incident as a whole. It's obviously very early stages at the moment."
Chelsea crashed out on penalties to the Championship side and, despite scoring his first goal of the season, Drogba's night was further soured after clashing with visiting supporters.
The 30-year-old's celebrations were cut short after scoring the opening goal in the 27th minute by several missiles thrown on to the pitch.
But Drogba, who gave the away fans a middle-finger salute, threw an object back and may now face an FA investigation and a possible misconduct charge.
Drogba, who was shown the yellow card for his actions, said: "This is something I want to make clear. I tried to celebrate the goal and I received some things at me.
"The big mistake I did was to throw it back so if someone was hurt, I just want to apologise for it.
"This is not something I should show in a football match and I want to apologise.
"It was an incident in the heat of the moment and I regret it. It was just a mistake and nothing more."
Drogba had put Chelsea ahead in their Carling Cup fourth-round tie with a superb curling effort into the bottom corner.
But a shock equaliser 20 minutes from time through Burnley's veteran striker Ade Akinbiyi stunned the Premier League side.
With the tie still level at 1-1 at the end of extra time, Burnley goalkeeper Brian Jensen, nicknamed 'The Beast' by his team-mates because of his size, saved from Wayne Bridge and John Mikel Obi to send Chelsea crashing out of the competition.
Chelsea will now await the referee's report before commenting further but Burnley manager Owen Coyle and Chelsea coach Luiz Felipe Scolari both denied seeing the incident.
Scolari said: "I have not spoken with Didier about any problem. The referee gave him a yellow card but I did not see what happened."