The police officer who was at the scene when verbal abuse was hurled at worshippers at the Al Noor Mosque is no longer working in Christchurch due to a formal complaint about the incident.
The Police Minister has called on the apparent Trump supporter to hand himself in.
The man is wanted by police after he reportedly shouted Islamophobic comments outside the Al Noor Mosque in Christchurch.
A social media post on a Christchurch page for members of the Muslim community showed police officers watching a man in a black T-shirt emblazoned with "Trump for New Zealand".
The post said the man was swearing and making anti-Muslim comments.
The incident occurred about 4.30pm yesterday at the mosque in Deans Avenue in central Christchurch where 42 people were killed during the terror attacks three weeks ago. The attacks also left eight others dead and dozens more injured.
"He was there for about 15 minutes, he also kicked and tried to damage some of the memorial items along Deans Ave in front of the masjid," the post said.
"The police officer who was there let him go (claiming that he can't arrest him because he didn't break any laws), and afterwards the superior officer came on the scene and said they were trying to find him to arrest him," the post said.
The police officer who failed to arrest him is no longer working in Christchurch due to a formal complaint about the incident.
After the incident the man is said to have walked across Hagley Park towards the city.
Complaint laid over police response
A man who witnessed the anti-Muslim tirade, Adrian, said he had laid an official complaint over the police letting him walk away.
He said the man had been saying all Muslims were terrorists and needed to get out of New Zealand.
The man was spoken to by armed officers guarding the mosque, but Adrian says he was surprised and disappointed to see him let go.
"So I approached an officer and I said to him 'what's happening with this guy, why are you not detaining him or anything?'
"And the officer said to me 'do you know what freedom of speech is, this is New Zealand mate'. And I said 'what?', and he said 'yeah, he can say whatever he wants'."
Police are investigating how the matter was handled. Police were also criticised about three weeks ago after deciding not to charge a man who stood outside a Palmerston North mosque wearing a swastika just three days after the attack.
Meanwhile the Christchurch CIB is appealing to anyone with information about the man to contact them.
Police described the abusive man as Caucasian with dark hair and of solid build. As well as the t-shirt, he was also wearing a dark baseball cap.
A spokesperson said officers were actively seeking the man, who could face a disorderly behaviour charge.
Minister urges abuser to surrender to police
Police Minister Stuart Nash criticised the man's behaviour.
"I'm not aware of the exact details, but what I do know is the police are asking for this man to come forward and my advice to the chap is to hand yourself in for goodness sake," he said.
"Go to the police station and get this thing sorted. But that sort of behaviour is totally unacceptable."
Canterbury District Commander Superintendent John Price said the incident was being treated "extremely seriously".
"I spent time at the Masjid Al Noor mosque last night and the community is understandably shaken by this behaviour.
"Two armed officers were outside the mosque at the time, we are reviewing our response to this incident.
"Our community has no tolerance for those who target or victimise others because of their identity, and nor does police. This is especially so for members of our Muslim community who are already dealing with so much."
Anyone who recognised the man or who was in the area last night and saw anything that might help was urged to contact Christchurch Police on (03) 363 2500 and ask to be put through to the CIB duty squad.
Information could also be provided anonymously via Crimestoppers on 0800 555 111.