A report from Dow Jones suggests that the French group has already decided to pull the plug on the Internet business, but that it is likely to live on under new management.
Carrefour is thought to have received six approaches so far for the business, but has declined them all on the grounds that none of them reflected what it considers Ooshop to be worth.
The chain operates its online shopping business in both France and Spain, but it is thought to have accumulated losses in excess of €17 million, despite sales of more than €50 million, the report suggests.
All the potential buyers have been financial rather than trade, according to sources close to the company, and it is still remains unclear what form Ooshop would take once it left the Carrefour fold. Would it, for example, continue to be the main online portal for Carrefour's grocery business, or would it focus on mainly non-food items?
France's retailers have had mixed results with their online operations. Carrefour's decision to scrap its Internet shop in both France and Spain echoes that of rival Auchan, which has already scrapped its online shop in Spain and restricts its French operations to the Paris region.
But Casino, another of France's main supermarket operators, has had a huge success with its online arm, Cdiscount.com, helped in no small part by the fact that it focuses entirely on non-food products, sold at a significant discount to traditional store-based operations.
And with Spanish retailer Caprabo generating more sales than ever through its online business, could it be that the real problem for both Auchan and Carrefour is that they simply have not found the right format - or rolled it out widely enough - to be successful?
Major retailers can generate significant business through the Internet. In the UK, online shopping is becoming increasingly popular, with Tesco and Sainsbury both operating highly successful home delivery businesses, while smaller supermarket chains such as Iceland or more specialist retailers such as Thorntons are also moving rapidly into cyberspace on the back of good initial results.
Is Carrefour being premature in its decision to scrap the online business, then? Almost certainly. As the level of interest in Ooshop shows, there is a growing feeling that the Internet represents the future of grocery shopping - or at least a significant additional source of revenue for retailers - in a world where consumers have less time (and inclination) to spend walking the aisles.