The man was reportedly captured on CCTV in Belfast boarding a flight to Edinburgh within 48 hours of the killing before travelling on to Eastern Europe.
Gardaí investigating the murder are keen to speak to him.
Mr Dunne’s body was found last Saturday afternoon at a home on Boreenmanna Rd, with his head and arms removed.
The man, who gardaí would like to interview, flew back to Eastern Europe last Sunday, a day after the body of the 64-year-old Churchfield native was discovered at Castlegreina House.
Gardaí are believed to be liaising with colleagues in the North and in Scotland and could issue a European arrest warrant if the man does not return to Ireland.
It is reported that the person of interest may have been occasionally squatting in Castlegreina House.
However, the man is believed to have pre-booked the flights before the murder, and did not flee from Cork afterwards.
Superintendent Michael Comyns said: “He has returned to Eastern Europe in a pre-planned trip. He is one of a number of people we want to talk to in relation to the incident.”
It is believed the man has also booked flights to return to Ireland. He has been living and working in Cork in recent years and is not homeless.
Gardaí have become aware from their investigations that the man had been at the location where Mr Dunne’s remains were found in the days around the murder.
Since it became vacant over a year ago, the area at Castlegreina House has been frequented by homeless people, and users of homeless services, including Mr Dunne.
Mr Dunne lived in nearby high-support accommodation provided by Cork Simon. He was last seen there on December 27.
Gardaí have refused to say how many “persons of interest” are being investigated as part of the probe into the death.
Meanwhile, Mr Dunne’s body is expected to be released to his family on Monday for his funeral.
A team of up to 50 gardaí are now working on the case.