Labour has called for a new inspection requirement for state schools to deliver a "broad and balanced" curriculum, in the wake of claims that Islamist extremists have tried to influence teaching in 21 schools in Birmingham.

The news came as the education secretary, Michael Gove, denied the suggestion that a public row between him and the home secretary, Theresa May, had led him to consider his position in the cabinet.

Asked if the "very public row" had damaged David Cameron's government and if he was considering his position in the light of Ofsted's findings, Gove responded: "No."

The education secretary's brief response came at a Policy Exchange event in central London as it was reported that a school mired in the Islamist controversy in Birmingham was told it faced being downgraded in a damning Ofsted report.

And he refused to expand on his one-word answer as he was pressed about the impact of the row over tackling extremism and slipped out of a side entrance at Microsoft's central London offices where the event was held, avoiding further questions from reporters.

Days ahead of the expected conclusion of an Ofsted inquiry into the disputed "Trojan horse" claims, the shadow education secretary, Tristram Hunt, accused Gove of allowing education to be "skewed", school staff subjected to gender discrimination, and children exposed to extremist views in Birmingham, and called on him to go to the Commons on Monday to explain his actions to MPs.

Hunt's accusation comes after it emerged that a controversial Ofsted report on Park View academy, the secondary school at the centre of the Trojan horse allegations, said the school has failed to adequately warn its pupils about extremism and that staff are intimidated by the school's leadership.

The report, ordered by Gove and leaked to the Guardian, will highlight that one of the reasons the school will be downgraded to inadequate was that teachers had not received enough training in the government's anti-extremism Prevent programme.

Ofsted's report, which is due to be published alongside reports into a further 20 Birmingham schools on Monday, will say: "The academy is not doing enough to keep students safe, including raising students' awareness of the risks of extremism."

Appearing alongside Gove at the Policy Exchange event , Hunt was expected to accuse the education secretary of "gross negligence" for failing to ensure local oversight of schools to ensure their study programme was "broad and balanced".

"I think the time is right for Ofsted to introduce a new inspection criteria to its inspection framework, so that no school can be judged as good or outstanding unless it is delivering a broad and balanced curriculum that will equip our children with the skills for success in the future," Hunt was expected to say.

He will add: "Michael Gove's gross negligence comes at the cost of our children's education. For years he has been warned that the lack of local oversight in our school system will damage education standards. We are now seeing the results: schooling skewed to prevent a broad and balanced education; gender discrimination among staff; children exposed to extremist views."

Hunt was also expected to accuse Gove of choosing not to act, despite being forewarned in 2010.

Park View is one of five mainly Muslim schools expected to receive Ofsted's lowest rating, following a series of inquiries that has prompted public criticism of Gove by May, and led to David Cameron ordering an inquiry into the political fallout.

The school rejected the findings of the Ofsted report as driven by media coverage of the Trojan horse letter, which alleged that there had been a longstanding attempt by conservative Islamists to hijack school governing bodies. There are now doubts that the letter itself is genuine.

Among Ofsted's recommendations are that the school's management must "urgently improve systems for safeguarding students", including greater use of the government's Prevent strategy, and secure "equality of opportunity for all staff and students".

The former leader of Respect and a former city councillor, Salma Yaqoob, said the "trickle" of leaked reports, including most recently that of Park View's, had "thoroughly damaged the reputation of Birmingham's schools and children" while there was still no evidence of a radical Muslim plot.

Yaqoob, who lives in Birmingham, claimed the Ofsted inspections "were not impartial" and that "the view on the ground is that this is political interference – Michael Gove is the education secretary and his views are well known".

She told Radio 4's Today programme: "It seems the kids of Birmingham are already damned as being extremists before the official Ofsted report.

"I'm a Birmingham mum, I don't want to see extremism in Birmingham, I don't want to see any extremism in schools yet I have still to find any shred of evidence what extremism plot was going on.

"How are the kids being radicalised, are the kids doing something dangerous? Yet instead we've had this political footballing going on."

The shadow home secretary, Yvette Cooper, claimed that Gove's public spat with May had exposed the government's failure to develop a proper anti-extremism strategy.

Cooper highlighted official figures showing spending on the Prevent initiative to counter radicalisation had fallen from £17m in 2010, to £1.7m last year, and funding for police work on the strategy decreased from £24m to £18.7m over the same period.