The two inquiries, announced by Home Secretary Theresa May, will investigate alleged child abuse by MPs and wider public institutions.
The leaking, from her department, of a letter to Michael Gove, raises serious questions for the Home Secretary to answer.
One school was praised by Ofsted for using money from the pupil premium to pay for a number of cultural trips to Saudi Arabia.
Education Secretary Michael Gove has apologised to Home Office counter-terror chief Charles Farr and to David Cameron over his row with Home Secretary Theresa May, whose special adviser Fiona Cunningham has resigned, Downing Street said.
Falling out with the Home Secretary was not exactly what you would call a good start to this new parliamentary session for Michael Gove
And next week there may be some very awkward questions for the Education Secretary when those reports are published into alleged Islamification in some Birmingham schools.
Nevertheless, he does have a lot of support from the Prime Minister, who is seen as a very close ally of Mr Gove, and I do not think there is any suggestion at the moment that his job is on the line.
The Department for Education has defended school inspections criteria against criticism from the opposition after students were allegedly exposed to extremism in Birmingham.
A spokeswoman for the department said: "All schools, including academies and free schools, are required to offer a broad and balanced curriculum.
"Ofsted's inspectors already consider how this is delivered and pupils' social, cultural, and moral development when deciding a school's rating."
Education secretary Michael Gove was asked if he had considered his position today, in the wake of a report into an educational trusts failure to protect pupils from risks of extremism.
The minister replied to ITV News Political correspondent Libby Wiener by saying "No."
The Policy Exchange event, in which Gove spoke of his plans to end illiteracy within a generation, was quickly wrapped up after the question regarding his position was asked.
Labour's shadow education secretary Tristram Hunt said the current inspection criteria was "not fit for purpose" and should be extended to ensure problems like the concerns around protecting children from extremism do not emerge again. Speaking to Radio 4's Today programme, he said:
I don't think the inspection criteria is fit for purpose, in the kind of schooling we want. We want a much broader criteria to ensure that we don't have these problems arise.
He also called for a local oversight of schools, with directors on the ground in charge of driving school standards in a bid to avoid the problems affecting schools in Birmingham.
You can't run all these schools from behind a desk in Whitehall so we need local systems of oversight and accountability.
We need a local director of school standards so we don't end up like the kind of situation we're seeing in Birmingham which is of national significance.
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 attempted to influence teaching in a set of schools in Birmingham.
Shadow education secretary Tristram Hunt accused Michael 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 come to the Commons on Monday to explain his actions to MPs.
The school at the centre of the row over alleged Muslim extremism in Birmingham has been criticised in an official report.
ITV News understands that an Ofsted investigation of Park View School found children were not protected from extremist views and that leadership was weak.
Prime Minister David Cameron has defended the government's handling of the affair, adding that he would be stepping in to "sort out" a dispute between the Home Secretary Theresa May and Education Secretary Michael Gove.
ITV News correspondent Rupert Evelyn reports:
ITV News understands that two Birmingham schools run by the same educational trust have been criticised by Ofsted for apparently failing to protect pupils from the risks of extremism.
Golden Hillock School and Park View Academy have both now been rated "inadequate" and placed into special measures.
The two schools are both run by Park View Educational Trust, which also manages a primary school.
A copy of the Ofsted report into Golden Hillock says "too little is done to keep students safe from the risks associated with extremist views".
ITV News understands that the report into Park View Academy (PVA) contains similar criticisms and that its acting principal, Monzoor Hussain, has now stepped down as acting headteacher.
It is also claimed staff at PVA have expressed concerns about both the governors and the leadership team and that they do not have faith in them.
Asked for a response to the claims, Park View Academy refused to comment.
One of the Birmingham schools at the centre of the 'Trojan Horse' investigation has issued a response to a critical Ofsted report, pointing out that the regulator makes no suggestion it tolerates extremism of any kind.
A statement from Park View Educational Trust, which runs Golden Hillock School, says: "Ofsted judges that Golden Hillock is not doing enough to raise students’ awareness of the ‘risks of extremism’.
However, it is crucial to note that the Ofsted reports make absolutely no suggestion, nor
did they find any evidence, that Golden Hillock either promotes or tolerates extremism or
ITV News has seen an Ofsted report that strongly criticises the governance of one of the Academy schools at the centre of the 'Trojan Horse' investigation into possible links with hard-line Islamists.
The says measures to keep students safe are "inadequate", while the school's equalities policy was deemed "not fit for purpose".
The regulator also warns that "leaders and governors are not doing enough to mitigate against cultural isolation", while female staff complained about feeling "intimidated" by male colleagues,
Some of the key points raised in the report:
These concerns, along with several problems with the teaching and curriculum, have led to the school being placed in 'special measures' by the regulator,
This means the school has to follow an action plan to improve its performance or risks being closed down.