Fox is reportedly one of four on a shortlist being considered by the broadcaster to replace Grade, who announced in April he will leave at the end of this year.
Others reportedly being considered are John Cresswell, currently chief operating officer of ITV, Tony Ball former Sky chief executive and Pascal Cagni, head of Apple Europe.
However, Fox is reportedly the front runner.
He has been widely credited with turning the entertainment retailer round since joining from Kesa Electricals three years ago.
The retailer’s sales were in decline when he joined in 2006 but grew 14% in the fourth quarter last year despite music and DVD retailers coming under pressure from internet retailers and illegal downloading.
Fox would inherit an even bigger challenge at ITV, which forecast net TV advertising revenue would decline 16% between April and June.