NSW Police said on Tuesday it is looking into the man's case.
"Officers from Tweed/Byron Police District, with assistance from Richmond Police District, have commenced an investigation following reports of potential breaches of Public Health Order. Inquiries are continuing," a spokesman told SBS News in a statement.
When asked about the Sydney man, NSW Health Minister Brad Hazzard said he would not expand on details given the police investigation.
"All I'm prepared to say about our traveller to Byron is the police are looking extremely closely what he was doing in that area," he told reporters in Sydney on Tuesday.
"I trust the police will be able to take appropriate action in due course."
An exemption to Greater Sydney’s current lockdown rules allows people to travel outside the area to “inspect a potential new place of residence”. It’s unclear whether the Sydney man relied on this to travel to the area.
Mr Hazzard said his legal department would be looking at whether rules around real estate viewing exemptions needed to be tightened. But he said he was concerned some people would try to breach rules regardless.
"What worries me is no matter what legal orders or requirements are in place, you can't legislate against stupidity, arrogance and entitlement," Mr Hazzard said.
There were long lines at a pop-up drive-through COVID testing site at the Cavanbah Sports Ground on Tuesday, after the region was sent into lockdown for at least one week.
Source: Byron Shire Council
The lockdown will remain in place until at least 12.01am on Tuesday, 17 August.
There were no new cases detected in the region on Tuesday. One case was recorded in the Northern Rivers which was not linked to the Sydney man.
Byron Shire Mayor Michael Lyon earlier told SBS News the Sydney man left his COVID-positive wife at a Sydney hospital before travelling with his teenage children to the Byron area.
Mr Lyon said the man was in the community looking for property for four days while potentially infectious.
He said the man did not use QR codes, making the task for contact tracers more difficult.
Mr Lyon said he was "disappointed".
“You half expect it in a pandemic, you know that these things can happen but when it's through sheer stupidity and negligence of someone involved, it just makes it really annoying and frustrating,” he told SBS News.
“So many peoples’ lives and livelihoods depend on this.”
The Sydney man is now at Lismore hospital.
Mayor slams 'negligence' of man who led to Byron Bay lockdown
NSW Health has advised anyone in Byron Shire, Richmond Valley, Tweed, Lismore, Kyogle and Ballina to immediately isolate and have a COVID-19 test if they have even the mildest of COVID symptoms.
More than 80 per cent of the NSW population is in lockdown as the state struggles to curb the spread of the Delta variant beyond Greater Sydney and into the regions.
About 6,571,800 residents have been placed into lockdown.