Two people have presented themselves to Goulburn Police Station after allegedly fleeing quarantine officers at Melbourne Airport on Friday.
Victorian Health Minister Martin Foley has promised fines of up to $19,000 each for the pair.
Addressing media on Saturday, Mr Foley described it as a very serious matter.
The man and woman flew from Canberra to Melbourne on Friday but they allegedly ran away and escaped in a waiting car after being told by authorities they would have to quarantine for 14 days due to coronavirus restrictions.
"They have now apparently returned to Goulburn, NSW," Mr Foley said.
"Acting upon the advice of the public health team, these people have breached numerous orders of the state of emergency in Victoria and each will be fined at least $19,000 for their breaches.
"This is a very serious matter. We've discussed the link between the NSW cluster and Victoria and we're determined to get on top of it. That requires what appear to be tough measures for some."
A Victoria Police spokeswoman told The Post that police and the Department of Health and Human Services would "jointly investigate all circumstances surrounding the incident and enforcement actions will be taken."
NSW Police Media spokeswoman said the 24-year-old woman and 26-year-old man presented themselves to Goulburn Police Station at 10am Saturday.
"They have been directed to self-isolate at their home in southern NSW while NSW Police seeks urgent advice from NSW Health," she said.
Officers will also perform checks to ensure compliance.
Victorian Police had been informed.
The Post has requested comment from the Southern NSW Local Health District on whether the couple have been tested for coronavirus.