She says some community leaders see schools as a way to "indoctrinate impressionable minds" and bring children up with narrow horizons.
Premier spoke to Simon Calvert from the Christian Institute, who were specifically referenced by Spielman as a group with "conservative" views.
Calvert said: "I don't quite know what she means - she was talking about voices within Anglicanism and the Christian Institute is not an Anglican organisation."
"We have of course disagreed with Ofsted in the past and we've been public in our criticism in some of the things they've got wrong, they've obviously remembered we've done that but I didn't think it was terribly relevant really for her to mention us in that setting."
Speaking of her call for "muscular liberalism" in the face of some religious schools promoting their views over others', Calvert said: "muscular liberalism shouldn't mean some kind of aggressive secularism - and that's the danger. Of course we would like to see Ofsted led in a way that does promote the great British tradition of genuine liberalism."
He added: "She doesn't seem to reflect a full understanding of what liberalism really is - which is freedom to disagree, freedom to have different points of view".
"I think a lot of Christian educationalists are concerned about the direction of travel...they are day-in day-out helping to run schools...they don't feel they get much credit for it they feel that instead senior educational figures talk about Christian education and religious education and religious schools as if they are the problem."
Click here to listen to the full interview with Simon Calvert speaking to Premier's Cara Bentley:
