Cameron wanted Fairhead to stay on as chair of the BBC Trust until 2018 to oversee the BBC’s transition to being governed by a new board that unified executives and non-executives.
However, The Guardian reports that Fairhead was told just after the parliamentary summer recess by Theresa May and the culture secretary Karen Bradley that the government wanted to reopen the appointment process and she would have to reapply.
Fairhead, the former chief executive of The Financial Times Group, said in her statement, initially given to the Financial Times: "I took on leadership of the trust to help stabilise, strengthen and develop the BBC following a very difficult period in its history and through charter review. I am proud of what I and my colleagues, both at the trust and the BBC, have accomplished during the past two years.
"The prime minister strongly encouraged me to take part in the new appointment process, for what would be a new four-year term as BBC chairman. However, after much thought I have come to the conclusion that I should not do so. It is my belief that it will be better to have a clean break and for the government to appoint someone new."
The culture, media and sport select committee criticised Cameron’s reappointment of Fairhead, saying there should have been an open selection process. She also lost a personal supporter when George Osborne left the Treasury.
Fairhead was first appointed to the role in September 2014.
