One source said Tamihere was challenged by police staff manning the southern border - but they reportedly allowed him through after he insisted they couldn't say no to him.
Tamihere told them that he could cross the border because he was an essential worker.
However, it is understood the trip was not considered official police business and police have confirmed no exemption was given for travel.
Due to several investigations into the allegations police could not comment on the specifics of the incident.
The Independent Police Conduct Authority has been notified and fellow officers are now investigating the border breach allegations.
Tamihere has been contacted but did not want to comment.
A police spokeswoman said the organisation was "very concerned about this reported incident".
"For this reason, police has self-referred this matter to the Independent Police Conduct Authority," she said this afternoon.
"An investigation of the alleged breach is being led by staff from another district, and so further comment is not appropriate at this time."
Earlier the spokeswoman confirmed the alleged offender crossed a "boundary checkpoint without an appropriate exemption".
"As police currently understand it, this involved a member of police accompanying a group of people across an alert level boundary so they could attend a burial a short distance away," she said.
"Police have confirmed the travel was not permitted ... but further inquiries into the matter are required to more fully understand the context, including decision-making around the case."
A source said Tamihere was well known and respected for his close connections and good engagement with local iwi.
They said it would be a shame if that work - and that of other police officers in similar liaison roles - was tarnished by the alleged incident.
The source hoped people would reserve judgment until all the facts were clear and refrain from criticising Iwi or Maori in any way.
Police Minister Poto Williams said if the allegations against Tamihere were true she would be "quite disappointed".
She said she had been given "very brief" information about the situation and was aware it was under active investigation.
"But should it be a case of somebody doing the wrong thing when everyone is doing the right thing, I'll be very disappointed," she told the Herald.
"I think we should all do what we're supposed to do."
Tamihere, a former Auckland Blues Super Rugby player, is one of four iwi liaison staff in Counties Manukau, one of 12 in the wider Tamaki Makaurau area and one of 56 across the country.
Each iwi liaison officer helps to "navigate cultural issues and work on improving police relationships with Māori".
The group is made up of police officers and police employees, based in every police district in the country.
• Giving advice on cultural issues, such as helping both sides understand the procedures and protocols of the other when a sudden death happens;
• Helping local police work with iwi and whanau on preventing crime, crashes and victimisation;
• Providing leadership for special events such as Waitangi Day and Māori cultural events;
• Dealing with the cultural aspects of major incidents like the Canterbury earthquakes; and
• Helping with major criminal investigations involving Māori.
Auckland has been in alert levels 3 and 4 - which carry strict travel restrictions - since it was announced in August that the highly infectious Delta variant of Covid-19 was present in the community.
More than 320,000 vehicles have been stopped at checkpoints on the city's southern and northern boundaries in that time, police noted this evening in the organisation's daily compliance update.
Of those, 5524 have been turned around.
Police have also noted over the past month multiple arrests for people crossing the border in breach of lockdown rules - the most notable being a couple accused of using their essential worker exemptions to drive to Hamilton before taking a flight to a holiday home in Wānaka.
In that case one of the alleged offenders was the son of an Auckland District Court judge.
"It's important to remember that travel across an alert level boundary remains restricted and you will be turned away if you don't have the required evidence for permitted travel, as outlined on the Covid-19 website," a police spokesperson said yesterday.
• A 53-year-old arrested at Wellington's ferry terminal after police said he crossed Cook Strait with a caravan he'd bought in Christchurch, after misusing work exemption to leave Auckland.
• A woman, 24, and man, 41, arrested in Wellington after allegedly travelling from Auckland.
• Two Aucklanders accused of using false documents to visit Taupō.
• Three essential workers from Auckland allegedly caught at Mt Ruapehu's Turoa Ski Field.
• A man who boasted on Tik Tok of making a cross-border McDonald's run.
• Two gang associates accused of trying to flee police on a gravel road near the lockdown border. The pair were later found to have over $100,000 in cash in the car and a boot full of KFC, police said.