According to a report in The Sunday Times, Tesco is keen to iron out recurring problems on its Tesco Direct online service, which has its first birthday this month, and will launch a 1,000-page searchable web directory for its non-food and drinks products this week.
It is understood that as part of the overhaul, Robinson, who is a former finance director at Argos, may step down after two year's service.
Tesco announced in March that it planned to restructure its marketing arm following the expansion of Tesco Direct, which retails electrical goods, home furnishings, DIY, furniture and sports accessories.
The £30m expansion, which increased the product range on Tesco Direct by 25% to around 7,000 products, led to the promotion of Kendra Banks to marketing director from head of marketing in March.
The supermarket giant said earlier this year that it planned to "raise the bar" in the home shopping sector by expanding Tesco Direct, which could put an additional £2bn on its annual UK sales.
Tesco has so far declined to comment on whether Robinson is to step down as chief executive, but City analysts have suggested he could make way as part of the development of the Tesco Direct online business.
The speculation about Robinson's future comes as rival Sainsbury's recently hired Tanya Lawler, a former Argos commercial director, as director of online.