However it is unclear whether any breach occurred and whether the pair would face legal action.
The man and woman left Auckland on Saturday.
It is understood they chartered from Skyline Aviation Limited in Hawke's Bay, which travelled to Auckland, collected them and proceeded directly to Wānaka.
Flight tracking data shows they landed in Wānaka just before midday.
Police were alerted and are now investigating.
"Police are aware of this matter and are making inquiries to establish if there has been any breach of the Health Order," said a spokeswoman.
"We won't be commenting further at this time."
The man is the founder of a company that deals in cryptocurrency investment and digital asset managment.
A source said the couple appeared to have all the relevant documentation that would allow them to leave Auckland and travel to Wānaka.
However police were waiting on confirmation from the Ministry of Business, Innovation and Employment.
The source urged people to remain calm until all the facts were established.
They said online abuse directed at the pair may be unfounded.
"People are sending death threats, it's disgusting," they said.
A lawyer acting for the entrepreneur, whom the Herald has chosen not to name at this stage, defended the travel.
He told Stuff his clients lived in Otago and their travel was for "legitimate business reasons".
An MBIE spokesperson said the entrepreneur and his employee held the relevant business travel documentation required to cross alert level boundaries.
However the circumstances of the trip were being reviewed.
"Only travel for the purpose outlined in the documentation is permitted," said the spokesperson.
"Any extra travel is an offence. The circumstances of the travel is being reviewed."
The pair's trip was a "completely different situation" to another case of Aucklanders leaving Level 4 and travelling to Wanaka, the source told the Herald.
William Willis, 35, whose mother is District Court Judge Mary-Beth Sharp, and lawyer Hannah Rawnsley, 26 were dobbed in after arriving at a holiday home in the southern resort town last week.
The couple left Auckland on Thursday, September 9, using essential worker exemptions to cross the boundary and drive to Hamilton Airport.
They then took a commercial flight to Queenstown via Wellington and hired a car to drive to Wānaka, police said.
A tipster alerted police through the Covid-19 online compliance tool and the couple were approached on Saturday afternoon, police said.
The couple have since returned to Auckland.
After abandoning a bid for name suppression the couple issued a statement apologising for their alleged breach.
Police have indicated Willis and Rawnsley will likely face charges - but there might be some delay in getting the case before the courts because of court restrictions resulting from the ongoing lockdown.
"The decision that we took to travel to Wānaka last week was completely irresponsible and inexcusable," the couple said.
"We are deeply sorry for our actions and would like to unreservedly apologise to the Wānaka community, and to all the people of Aotearoa New Zealand, for what we did.
"We can confirm that as part of routine testing for essential workers when crossing the Auckland border, we both received negative Covid-19 tests prior to undertaking the travel, and on our subsequent return to Auckland. We can also confirm we were not considered close contacts nor had we had visited any locations of interest.
"We understand that strict compliance is required to stamp out Covid-19 from our country. We have let everyone down with our actions, and we wholeheartedly apologise."
Auckland remains under alert level 4 - the most stringent Covid-19 lockdown - until at least next week, while the rest of New Zealand is in alert level 2.
