Victorian Chief Health Officer Brett Sutton said a woman tested positive after attending the rally but argued it was very unlikely she would have been infected at the protest.

Prof Sutton said she had worn protective equipment and had mild symptoms, making it unlikely to have transmitted the virus.

"Because we've prompted everyone who's attended the protest with any symptoms...to come forward for testing, this individual has been identified as a positive case," he told reporters on Monday.

Encouraging protesters to undergo testing if they developed any symptoms, Prof Sutton said he wasn't convinced there had been transmission at the protest.

"It's really in that first week that people overwhelmingly develop symptoms, so any cases really should be out there now," he said.

The protester was one of Victoria's 12 new coronavirus cases recorded on Monday, seven of which were linked to a family outbreak that has seen 11 members infected.

The extended family has had gatherings across different households in Coburg, Broadmeadows and Pakenham.

IN OTHER NEWS

Four of the family members to test positive are children who attended two different schools in Melbourne, prompting their closure while cleaning and contact tracing are carried out.

Pakenham Springs Primary School will be closed for at least one day from Monday.

St Dominic's Primary School in Broadmeadows will also be closed while testing of students and teachers that attended the school from May 26 to June 3 happens.

Investigations are underway to identify whether any staff or students must self- isolate, the education department said.

"No fault of that family," Prof Sutton told reporters.

"If any of us are unwell, if any of our kids are unwell, even with mild symptoms, then we need to exclude them from seeing others, including scheduled gatherings."

Two other cases recorded on Monday were linked to a patient of Monash Health who tested positive on Sunday.

Although Victoria reached double-digit cases on Monday, Prof Sutton remained confident of the easing of restrictions scheduled for next week.

Cafes, restaurants and pubs will be able to increase their capacity from 20 to 50 patrons from 11:59pm on Sunday.

Patrons will be able to order a drink without buying a meal, but they will need to be served at a table, rather than the bar.

Non-contact sport will return for all age groups from next Monday, as will skiing, with the season to start from that date.

About 600,000 Victorians have been tested for the virus.

Between Thursday and Sunday, police did 1689 spot checks at homes, businesses and non-essential services across the state.

This brings the total checks to 73,652 spot checks since March 21.

Only one fine has been issued since Sunday, to one of the organisers of the Black Lives Matter rally earlier this month.

- Australian Associated Press

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