Mr Norman said the departure followed the promotion to deputy chief executive of Allan Leighton, the former operations director, in April.
"We had a young guy who took over the marketing role for a while and he decided to leave," he said. "He has my full support. He wanted to get back down south and I am not sure he wants to stay in marketing. He goes with my best wishes."
There were suggestions that the boardroom changes meant that there was not enough room for a marketing director, but Mr Norman denied there had been any animosity over Mr Fleming's departure. Mr Fleming joined the Leeds-based group from rival Somerfield in September and left earlier this month with what Mr Norman described as a "modest" pay-off.
Mr Norman has previously undertaken the marketing role for a period of nine months, but is not likely to occupy the position permanently. "The position at Asda is that we change roles around from time to time. It is likely in the next six months that someone else will want to have another go at it," Mr Norman said.
He has been keen to hammer home the message of Asda as a lower-price alternative to rivals such as Sainsbury and Tesco. Since his arrival more than three years ago, he has reintroduced the old "Asda price" advertising slogan, emphasising value for money, while campaigns from rivals Sainsbury and Tesco have concentrated more on customer service.
His background includes a masters in business administration from Harvard University and a stint at management consultants McKinsey, but he rose to prominence as finance director of Kingfisher, the Woolworths to Comet retail group, before joining Asda.
Recently, he has been at the forefront of those who have attacked the move by the Chancellor, Kenneth Clarke, to extend income tax to cover share options, claiming it would hit the company's attempts to encourage a share-owning workforce.