“Two Oxfordshire schools run by the same [Catholic] group have been criticised for not being safe enough.
In their last inspections St Gregory the Great School and St Joseph’s Catholic Primary School, which both belong to The Dominic Barberi Multi Academy Company, were rated inadequate.
Executive principal Fraser Long said the schools were now “moving forward rapidly”.
But councillor John Howson called on the Education Secretary to intervene.
The latest Ofsted report for St Joseph’s Primary School in Carterton, published on Tuesday, said staff did not follow procedures to keep children safe.
An Ofsted report published in May describes staff and parents at St Gregory the Great School, based in east Oxford, as “rightly concerned” about safety, with pupils worried about poor behaviour.
The principal and vice principal of the school, which is in special measures, are on leave.
Mr Howson, who is a Liberal Democrat spokesman for education on Oxfordshire County Council, said the new Education Secretary, Damian Hinds, had to act.
“The present situation cannot be allowed to continue,” he added.
Anneliese Dodds, Labour MP for Oxford East, said concerned parents had contacted her about St Gregory.
She added: “The most important thing is that there is a clear plan to sort out these problems, and that there’s going to be some openness.”
Mr Long said “robust and swift action” was being taken on safeguarding.
He would not comment on why the head and deputy head were on leave at St Gregory but insisted their responsibilities were being covered.
The Department of Education said: “We are working closely with the Multi-Academy Trust to deliver the improvements this school needs and its pupils deserve.
“We take the safeguarding of pupils very seriously and where Ofsted finds failings on this, action will be taken, as we are doing in this case.””