He said detectives from both the homicide squad and fugitive taskforce, along with a number of other police units, had conducted a significant search for Mr Barbaro over the past three days, but had been unsuccessful.

"At this stage we are looking in all states of Australia and interstate police are assisting us," he said.

Mr Barbaro is a scion of the Barbaro crime family, an Australian branch of the mafia, and brother of slain Sydney underworld boss Pasquale Timothy Barbaro, 35, who was gunned down in a 2016 gangland hit.

The family declined to comment when contacted by The Age on Thursday morning.

Detectives spent Wednesday night negotiating with Mr Barbaro's family, visiting the home of his father, Giuseppe "Joe" Barbaro, in Melbourne's west in the hope of arranging the 33-year-old's surrender.

"We’ve relied on our own methodologies and inquiries to locate him, and 99 times out of 100 they're successful, but at this stage we have not been able to locate him," Detective Inspector Day said on Thursday.

The senior officer urged anyone who may have seen suspicious activity in South Melbourne's Park and Ferrars streets during the past fortnight to contact police.

"We are looking for information from residents, and people in that area, during the two weeks leading up to April 29," he said.

"In particular the afternoon of Tuesday the 28th and into the early hours of Wednesday the 29th last week."

Police began searching for Mr Barbaro on Tuesday in the hope of questioning the Queenslander after it was revealed he'd been in a casual relationship with Ms Price, a mother of one, since about October 2019.

Police had been called to Ms Price's two-storey Park Street townhouse, at the rear of a housing complex, after her family reported losing contact with the 26-year-old on Monday. Inside, they found her body.

Detectives said she'd suffered a "significant assault".

Forensic analysts spent more than 24 hours inside trying to piece together what had occurred. Neighbours were told to hand over any CCTV camera footage they may have had from the previous six days.

Residents said their pocket of the suburb, on the western side of Ferrars Street, mainly housed families and the elderly and was usually free from crime.

Police sources told The Age Ms Price may have been dead for a number of days before being found.

The Barbaro family, with known links to the Calabrian mafia, are no strangers to the spotlight.

Another of Mr Barbaro's brothers, Queensland man Rossario Dom Barbaro, took his own life at the age of 28, just months after his older sibling's 2016 murder.

Harley Barbaro, a 27-year-old Gold Coast man and alleged Villains gang member, is another brother.

All four are sons of Joe Barbaro, a cousin of gangland figure Pasquale Barbaro, who was gunned down alongside Jason Moran in a van while watching a children's football training session in Essendon in 2003.

Joe Barbaro was jailed in 2005 for six years for his part in an illicit drug ring in Melbourne and Sydney.

His father and his son, both also named Pasquale, were killed in gangland hits – his father in Brisbane in 1990.

In 2004 Joe Barbaro’s baby daughter Montana, then three weeks old, was snatched from her car seat in a Brimbank shopping centre during a brazen kidnapping.

She was found two days later with a shaved head in a derelict house in North Melbourne, after a passerby heard her cries.

In 2012 another related Pasquale Barbaro, the son of Joe’s uncle, was sentenced to life in prison for importing what was then the world's biggest ecstasy haul into Melbourne.

On Thursday, Detective Inspector Day said police knew the wanted man had been in touch with family and friends. He warned those people that anyone caught harbouring a fugitive or assisting in their escape could face serious criminal charges.

"Barbaro, this is not going to go away. This is an opportunity for you to come forward," Detective Inspector Day said.

"We have a young female, 26 years of age, who has died prematurely.

"She has a family who is grieving for her and a community that is needing and deserving of answers."

Ricardo Barbaro is described as being 185 centimetres tall with a solid build, black hair and an olive complexion.

Police said he is known to frequent the north-west and western suburbs of Melbourne.

Detectives are also searching for his white 2009 Toyota Hiace van with registration 1OZ 8PC.

The victim’s vehicle, a white 2017 E350 Mercedes sedan with distinctive pink numberplates 22ZERO, is missing from her property.

Police have released an image of Ricardo Barbaro and a picture similar to Ms Price's vehicle in the hope someone can provide information regarding his whereabouts.

Anyone with information on the South Melbourne homicide investigation is urged to contact Crime Stoppers on 1800 333 000 or visit crimestoppers.com.au.