Sharon Evans, who also runs the charity Dot Com, told John Nicolson the child abuse inquiry was all about controlling information, and she fears the Grenfell Tower inquiry will proceed along the same lines.
Evans was a senior figure and a public face at the inception of the inquiry, which was set up in 2015 following the Jimmy Savile paedophilia scandal while May was still home secretary.
She told John Nicolson that she soon began to harbour concerns about how the inquiry was being run, and these concerns prompted her to suggest to her fellow panellists that they write to Theresa May.
She continued: "I was taken to one side and it was made clear to me, I was told that Theresa May was going to be the Prime Minister, this inquiry was going to be part of this, and that if I didn't toe the line and do I was told, if I tried to get information out, I would be discredited by her advisors."
She went on to allege that this warning was issued by the inquiry's QC, Ben Emmerson, who served as a go-between between May and the inquiry team.
Evans also claimed that, when talking to the Home Affairs Select Committee, she was given a 23-page document of what she should say and told then-chair Keith Vaz "I fear that I may not be able to tell the truth because I have been told what I may and may not say."
Although Evans and her colleagues were promised the inquiry would be independent, she told John "there was no independence" and the confidentiality clause imposed on all panel members facilitated the "suppression" of the truth.
When asked who wanted to suppress the information, she said "my belief is that it was the Home Office."
Grenfell, she continued, looks like it will be "exactly the same... I want to try to stop them going down the same road, where they are controlled and information is suppressed. When there is an inquiry they can gather all the information and then they can choose what is released. That is what I think is extremely dangerous."
An IICSA spokesperson said: "We are not aware of any such conversation.
"As we noted yesterday, this Inquiry was established in March 2015. It is a statutory inquiry and our independence is underpinned by the Inquiries Act 2005. This is different from the previous non-statutory process."
Listen to the full interview above