The former Financial Times Group chief executive is the front-runner to replace Lord Patten, who stood down in May for health reasons after a turbulent three years in the job.
The BBC reported on its website that Ms Fairhead had been appointed to the role, and in doing so would become the first woman to chair the BBC Trust. But a statement from the Department of Culture, Media and Sport described her only as the Government's "preferred candidate".
Ms Fairhead will appear before the Culture, Media and Sport Select Committee for pre-appointment scrutiny on September 9.
Culture Secretary Sajid Javid said: "Rona Fairhead is an exceptional individual with a highly impressive career history. " Ms Fairhead said: "The BBC is a great British institution packed with talented people and I am honoured to have the opportunity to be the chairman of the BBC Trust."
Education Secretary and minister for women and equalities, Nicky Morgan, said: "This is a milestone moment and comes at a time when there are more women at work than ever before, represented at higher levels."