The Victorian premier has said she expects police to take action over the “utterly unacceptable” display of Hezbollah flags at weekend protests in Melbourne against Israel’s attacks on Lebanon.

Jacinta Allan said on Monday it was her “expectation” that “agencies pursue this” despite the AFP warning the display of terrorist symbols must meet particular conditions to warrant a charge.

“This is driving deep grief and division here on the streets of Melbourne,” she said. “I expect the police agencies to pursue this.”

Her comments came after the Australian federal police indicated they had powers to ask people to remove symbols prohibited under hate speech laws, which include symbols of terrorist organisations, but not to physically remove them directly.

Allan said Victoria police were working with the AFP to hold people displaying the symbols to account. The flag is considered a prohibited symbol as Australia has designated Hezbollah a terrorist organisation.

“Bringing grief and pain and division to the streets of Melbourne by displaying these prohibited symbols, is utterly unacceptable,” she said.

Allan said the state government stood with Victoria’s Jewish community “at this incredibly difficult time”.

The federal government is threatening to cancel the visa of anyone inciting “discord” in Australia and has warned against importing “radical ideologies of conflict”, after weekend protests against Israel’s attacks on Lebanon.

Some marchers carried Hezbollah flags and portraits of leader Hassan Nasrallah, who was killed in an Israeli attack.

NSW police allegedly pepper sprayed a 13-year old child and two others after officers attempted to pull a Hezbollah flag off a protesters at the Palestine-Lebanon rally in Sydney, according to legal observers.

Legal observers at the rally on Sunday said police deployed pepper spray “indiscriminately” on the crowd after the protester did not surrender the flag.

The home affairs minister, Tony Burke, issued a statement on Monday saying “any indication of support for a terrorist organisation is unequivocally condemned”.

“It draws the immediate attention of our security agencies,” Burke said. “There is a higher level of scrutiny if anyone is on a visa. I have made clear from day one that I will consider refusing and cancelling visas for anyone who seeks to incite discord in Australia.”

The opposition has speculated that people carrying photographs of Nasrallah at the rallies are visa-holders and called for visa cancellations. No evidence has been provided to substantiate the claim that those carrying the photographs are visa-holders and not Australian citizens.

Anthony Albanese followed Burke’s statement with his own, issued ahead of Monday morning’s cabinet meeting. In brief public remarks ahead of the meeting, the prime minister spoke mostly about the economy but added some remarks about the Middle East.

“We’ve seen worrying signs over the weekend,” he said. “We do not want people to bring radical ideologies of conflict here. Multiculturalism and social cohesion cannot be taken for granted and it’s important that we continue to stress that that is the case as we go forward as well.”

Peter Dutton suggested protesters were praising Hezbollah leader Nasrallah and that some were visa-holders. The opposition leader did not provide any evidence to support the allegation.

“There are people who are living in the Jewish community in fear and there is an absolute outrage in relation to the glorification of a terrorist leader, which surely must be against the Australian law,” Dutton – a former home affairs minister – told Brisbane radio 4BC. “And if it’s not, the parliament should be recalled to pass a law that prohibits that from happening.”

Dutton said existing laws should be enforced.

“I find it completely unacceptable that the government wouldn’t be arresting people already or cancelling visas of people who are glorifying Hezbollah and Hamas and others. They have no place in our country.”

On Sunday, the Australia Palestine Advocacy Network held a national day of action for Gaza, staging rallies across the country.

Pro-Palestinian demonstrators gathered on Sunday at the state library in Melbourne and at Sydney town hall, and in smaller cities and towns, to protest against the escalation of the conflict.

The treasurer, Jim Chalmers, condemned anyone supporting Hezbollah or Hamas.

“Any indication of support for a terrorist organisation is completely and utterly unacceptable whether it’s Hamas or Hezbollah,” Chalmers said. “These are listed terrorist organisations and I think any support for a terrorist organisation should and will immediately draw the attention of our security agencies. There is a higher level of scrutiny for people who are on a visa.”

The Asio chief, Mike Burgess, said earlier this year “rhetorical” support would not automatically disqualify someone from holding a visa but support for violent extremist ideology would.

Hate speech laws which took effect earlier this year prevent the display of prohibited Nazi and terrorist symbols.

An Australian federal police spokesperson said that to be considered an offence, a prohibited symbol must be displayed in circumstances where the conduct involves “spreading ideas based on racial superiority or hatred; inciting others to offend or intimidate a person; advocating hatred of a person; advocating inciting others to offend; intimidating or using force or violence against a person or group based upon their race, religion or nationality”.

If the prohibited symbol is likely to offend, insult or intimidate people because of a defining characteristic it contained, it may also be an offence.

“Police officers have the power to direct a person to remove a prohibited symbol from public display, but they do not have the power to remove the prohibited symbol from public display themselves,” the spokesperson said.

A spokesperson for the attorney general, Mark Dreyfus, said: “It’s a matter for police to determine whether any action should be taken against those involved in the demonstrations.”