Senior Coalition women have rebuked Jacinta Nampijinpa Price’s comments about abortion, saying the Liberal party had “no interest in unwinding women’s reproductive rights” and saying it was an issue advanced by “fringe” politicians.

Even staunchly “pro-life” colleagues of the senator would not back her comments on Wednesday, with Queensland’s Matt Canavan saying it would not be helpful to discuss abortion in the lead-up to the federal election, and calling to “turn the temperature down” on the debate.

Sussan Ley, Jane Hume and Bridget McKenzie – three of the most senior female members of the Liberal and National parties – all flatly rejected putting abortion on the national agenda, saying it was an issue for states and territories. Hume, the shadow finance spokesperson, said: “A Dutton-led Coalition government has no plans, no policy and no interest in unwinding women’s reproductive rights.”

Price – the shadow Indigenous Australians spokesperson – told the Nine newspapers she “cannot agree” with later term abortions, which she claimed was “anywhere past the [first] trimester as far as I’m concerned … Full-term becomes infanticide.”

Later term abortion is a non-medical term for terminations after 20 weeks’ gestation, according to the Royal Australian and New Zealand College of Obstetricians and Gynaecologists, and can happen because of foetal abnormalities or if the mother’s life is at risk. The first trimester is 12 weeks.

Guardian Australia understands Price’s comments were not authorised by the Coalition leadership and are seen among the opposition’s senior ranks as unhelpful, ranging outside her own portfolio and risking a divisive and potentially damaging debate.

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It comes as several states debate abortion issues, sparked by conservative politicians seeking to re-examine the issue: abortion is featuring heavily in the Queensland state election; and South Australia’s parliament narrowly rejected a motion last week which would have wound back abortion care.

Ley, the deputy Liberal leader and shadow spokesperson for women, twice said on Wednesday the Liberals were not planning to address abortion laws.

“It’s important to remember that access to abortion is a state issue, which is why you often see it debated at state level, and we have no intention to change the settings from a federal health perspective,” she told Sky News.

“Obviously, individuals have their own views, and Jacinta is entitled, as a member of the National party, to her own view, but the federal Liberals have no intention of changing the settings when it comes to this issue.”

Hume told Sky: “It’s not going to happen.

“It’s not an issue for the federal government. In the Liberal party, it’s always been an issue of conscience, too, and rightly so. There are some deeply held views right around the country and that is fine. That’s why they call it choice.”

McKenzie told ABC Radio National that she accepted Price had strong views on the issue, but reiterated that it was not a federal issue. Asked for her views on the matter, McKenzie said: “I don’t think it’s helpful.”

Canavan, who has described himself as “pro-life”, said “I don’t think this is an issue best prosecuted through the heat of an election campaign.”

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“I don’t think there’s anything to be gained by playing it out in a tit-for-tat type process,” he told a press conference on Wednesday.

“It’s a really sensitive issue for people and I really feel for young mums who have been through this … I’m not ever going to have to experience that, I don’t think that says you can’t have a view on it, but I do try and be sensitive about it, but I’m not the one trying to raise this.”

The federal health minister, Mark Butler, also stressed it was a state responsibility, and noted the Labor government had improved access to non-surgical abortions by removing restrictions.

“I am committed to ensuring that all Australian women have equitable access to the healthcare they need, particularly women in rural and regional areas and those experiencing disadvantage,” he said in a statement.

But Larissa Waters, the Greens spokesperson on women, said she was “appalled” that politicians were debating women’s right to choose.

“Abortion is healthcare, and that’s a national issue, so what we should be talking about is how to make abortions affordable and accessible for everyone – not restricting the legal right to choose,” she said.

“It’s ludicrous that anyone would propose forcing someone to go through with an unplanned pregnancy against their wishes.

“If you don’t want to have an abortion, don’t have one, but you don’t get to tell other women what choices they should have.”