The Manchester United winger's £180,000 Bentley was hit during the incident, which left the Portuguese star shocked but unharmed
The 26-year-old player was not physically harmed in the crash, but witnesses claimed the accident left him "dazed".
A Greater Manchester Police spokeswoman said: "At about 6.40pm on Friday 15 February 2013, an unmarked police car responding to an emergency call was travelling along Gatley Road at the junction with Kingsway, Stockport.
"As the police car entered the junction it collided with a Bentley, Continental.
"The passenger in the police car received minor injuries and only minor damage was caused to both vehicles.
"The police driver has been suspended from driving duties in line with normal procedure."
A witness told the Manchester Evening News: "I could see the Bentley had gone head-on into the bollards at the traffic lights and the car had been a bit bashed up at the front.
"I wondered who was driving the Bentley and when I had a quick look in the police car Nani was there sat in all his United training gear.
"He looked a bit shocked and it had created a bit of a scene as it is a very busy junction and there were three police cars there dealing with the incident."