Police have said they are investigating newspaper reports that Cole shot a 21-year-old man, who was hit by lead pellets fired from an air rifle at the ground in Cobham, southwest of London, last week.
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Ancelotti, speaking before Chelsea host Premier League leaders Manchester United on Tuesday, declined to say what action Chelsea had taken against the England international, but told reporters: "Obviously, we are not happy with what happened.
"But I have spoken with him. He was very disappointed with this, said sorry, made a mistake. It was an accident.
"We have taken proper action but tomorrow he will play.
"The mistake was that the gun was here in Cobham. We didn't know the gun was here."
Asked to whom Cole had apologised, Ancelotti replied: "First, he said sorry to the guy that was involved in this, second sorry to the team mates, third sorry to the club."
He said the club had not considered sacking the player, something that would probably have happened in any other job.
"He is our player, he always had very good behaviour here," he said. "Now, we have to move on. Who didn't make a mistake in his life?"
A Surrey police spokeswoman said earlier that while no direct allegation had been made, the matter had been brought to their attention through media reports.
"We will be contacting the club in due course to establish whether any criminal offence has been committed," she said.
Media reports said the 30-year-old defender had mistakenly shot the man using a .22 air rifle, the most powerful gun available without a licence in Britain.
One eyewitness was quoted as saying the victim had "bled profusely."
Club staff treated the man, identified as sports sciences student Tom Cowan who was on work placement with Chelsea, according to reports. He did not need hospital treatment.
