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Premier Daniel Andrews on Tuesday said the new deaths were two people in their 90s, three aged in their 80s and one person in their 70s. Four of the six are linked to aged care facilities.
The deaths take the state's toll to 83 and the national figure to 167.
On Monday, Australia's second most populous state reported a record 532 new cases, but Tuesday's figures are still at levels alarming health officials.
Prime Minister Scott Morrison cut short a planned interstate tour to focus on the country's coronavirus crisis on Tuesday, citing a "very complex" outbreak in aged care homes in the city of Melbourne.
Premier Andrews announced that all non-urgent category two elective surgery will be suspended in metropolitan Melbourne to free up hospital space for at-risk aged care residents.
"I cannot stand here and tell you that I have confidence that staff and management across a number of private sector aged care facilities are able to provide the care that is appropriate to keep their residents safe," he told reporters.
There are currently 80 outbreaks and 764 active cases of the virus in private aged care residences.
Chief Health Officer Brett Sutton said there were 88 cases now at Estia Health Aged Care in Ardeer, 86 cases linked to some Basil's Homes for the Aged in Faulkner and 82 cases linked associated with Epping Gardens.
Seventy-six have been linked to Kirkbrae Presbyterian Homes in Kilsyth, 62 cases linked to Menarock Life in Essendon and 53 connected to Glendale Aged Care in Werribee.
"Some of the stories we've heard, some of what's gone on in some of these settings is simply not acceptable and it's not about blame, it's not about demarcations and having disputes about who is in charge of what," Mr Andrews said.
Aged care is the responsibility of the federal government and it has increased its interventions in recent weeks as deaths in Victoria's homes mounted.
"The commonwealth government - I've spoken with the prime minister, ministers have been speaking, officials have been speaking - it was a very long night, but this is a critical challenge," Mr Andrews said.
"This does not mean that every resident in an aged care facility, a private sector aged care facility that has an outbreak, will be moved en masse to a public, or indeed to a private, hospital.
"Where there is no confidence in infection control, where there is no confidence that care can be provided to a suitable standard, then we will do everything we can to move those residents out."
Health Minister Jenny Mikakos said the numbers in aged care facilities were distressing, citing St Basil's in Fawkner as a residence of particular concern.
Meanwhile, most of the tests for babies in the Royal Children's Hospital's neonatal intensive care unit have come back negative after an outbreak there, Ms Mikakos said.
The hospital confirmed on Monday it had one patient, a staff member and two parents who tested positive to Covid-19 in the neonatal intensive care unit.
"The latest advice that I have is that all the results have been received bar one result that's still pending of the babies at the NICU and they have all come through as negative," she said.
COSTLY MACCA'S RUN
A Macca's run has cost three mates almost $A5000 ($NZ5300) in fines for breaching Victoria's coronavirus stay-at-home orders amid the deadly pandemic.
Police handed out 79 fines in the past 24 hours to individuals caught disobeying Chief Health Officer Brett Sutton's directions to behave during COVID-19, the force confirmed on Tuesday.
Among those Victorians trying to dodge the law, three people from different homes were caught driving together to McDonald's after hanging out at a house.
Another seven people were fined at a short-term rental apartment in Melbourne's CBD, after having plans to party there for two nights.
Two Frankston residents were fined for staying in Murtoa, more than 300km from home, with no valid reason for travel.
Of the 79 fines issued, 23 were for not wearing a face mask in public.
Five fines were issued at vehicle checkpoints out of the 21,481 vehicles inspected on main arterial roads in the state.
Police conducted 4618 spot checks at homes, businesses and public places in the past 24 hours, boosting the total checks since March 21 to 175,472.
- AAP and Reuters