The chairman of the Northland-based iwi said it was important to note that no charges had been laid. He would not comment further.
The rare bird is protected and Mr Tau could face a fine of up to $100,000 if convicted.
Earlier today, MPs at Parliament refused to discuss the potential ramifications of the investigation for the leader of New Zealand's biggest iwi.
New Zealand First leader Winston Peters said if the allegation was proven there would be "serious consequences" but he did not want to "jump to conclusions without a court case".
Asked whether he had eaten the bird himself, the Northland MP said: "To the best of my knowledge, I don't know, but I'm not saying no.
"I think one time I was at a marae, at a place in the Hokianga, and I suspected it might have been but I didn't know it."
Maori Party co-leader Te Ururoa Flavell said he was concerned about any illegal act, but all the details of the investigation were not yet known.
"We always say that we leave it for our own people to make judgements on ourselves, so we have to leave it for that course to follow."
Hunting kereru has been illegal for nearly 100 years, though some Maori claim they have a traditional right to hunt the pigeon.
Asked whether protection laws were out of step with the Treaty of Waitangi, Mr Flavell said "The law is the law." He also said he had never eaten the bird.
Treaty Negotiations Minister Chris Finlayson refused to comment on any potential implications for Crown negotiations with Ngapuhi.
"Read my lips - I'm not commenting," he said.