The Greens may support the Coalition’s bill to limit gambling advertisements, potentially teaming up again after blocking Labor’s housing bill earlier this week.

The government is planning its own action on wagering ads with a parliamentary inquiry into gambling harms set to report back imminently.

But the Greens have challenged Labor to move faster, and Senator Sarah Hanson-Young has said her party is “sympathetic to any bill” to address the issue.

“The ball is in the government’s court to commit to stronger laws immediately and stop taking donations from gambling corporations,” she said.

After Opposition Leader Peter Dutton’s budget reply commitment to banning gambling ads during live sport, the Coalition introduced a private senator’s bill to that effect. Senator Sarah Henderson said it was time to “draw a line in the sand” on the issue, with the bill to restrict wagering ads an hour before and after matches.

“The community are over this. We in the Coalition are over this,” she said on Wednesday. “We are not going to wait for months, as the government has been signalling.”

The Coalition, Greens and numerous independents have called for immediate action on gambling ads, criticising the government for not moving quicker. Zoe Daniel and Rebekha Sharkie have introduced their own private members’ bills in the lower house, and David Pocock has moved stronger amendments to the Coalition’s bill in the Senate.

• Sign up for Guardian Australia’s free morning and afternoonemail newsletters for your daily news roundup

Hanson-Young called the Coalition’s bill a start, but said it was “not good enough”, saying the Greens wanted to ban all gambling ads on TV.

“The gambling industry feeds off addiction and profits from wrecking families, lives and sport,” she told Guardian Australia.

“The Greens have long called for a crackdown on gambling ads, donations and dinners and we are sympathetic to any bill which stops this cancer growing.”

The Greens and Coalition combining forces would create a Senate majority to pass the bill, as the unlikely pairing did earlier this week to overrule the government and defer the Housing Australia Future Fund bill until October.

The prime minister, Anthony Albanese, has expressed a personal dislike for gambling advertising but the government says it is waiting for a parliamentary inquiry into online gambling before introducing its own reform. The communications minister, Michelle Rowland, acknowledged to Guardian Australia “the status quo isn’t good enough”.

A spokesperson for Rowland said the government was “committed to strong consumer protections for online gambling”, and that the report of the parliamentary inquiry, chaired by the Labor MP Peta Murphy, would inform the government’s next moves.

“Any reforms in this area must be evidence-based,” the spokesperson said.

But gambling harm advocates expressed reservations about the Coalition’s bill, raising concern the restrictions may see gambling ads concentrated in other parts of the day, or on other mediums.

Tim Costello, the chief advocate for the Alliance for Gambling Reform, said wagering companies would “just work around it”, and was critical the Coalition had not taken stronger action when in government.

“They’ve thrown a better option to the public, but it’s still a sop to the sports betting companies,” he said.

“If the Coalition really want to reclaim the family friendly party, we know what parents want.”

The Coalition bill may return for debate on Thursday, the last parliamentary sitting day until August. It is unclear when the government may reveal its plans for tackling wagering harms, but Murphy said her committee’s report was “imminent”.

She said she was not able to detail exactly what the report would recommend, but said it would be a “thorough piece of work” going beyond just advertisements – and would feature an examination of social harms and education.

Murphy noted more than 150 submissions to the inquiry, which held a dozen public hearings.

A spokesperson for Responsible Wagering Australia, the peak body representing major online betting companies, acknowledged “community sentiment” to address advertising – but called for a suite of measures across all platforms, not just a “blanket ban on broadcast”.

The shadow communications minister, David Coleman, said the government would have to “explain to Australian families” if it did not back the Coalition bill.

“It makes sense for all parties to come together to support this commonsense initiative from the Coalition,” he said on Tuesday.