Concerns about a rapidly-spreading variant of Covid-19 prompted France to close its borders to the UK.
The store could run out of products such as lettuce, salad leaves and cauliflower if the ban continues, Sky News reports.
However, there are no concerns over Christmas produts as the needed items are already fully stocked.
Hauliers have not been banned from entering the UK but it is feared they will not want to enter the county if they cannot return to France.
Sainsbury's told Sky News: "All products for the Great British Christmas lunch are already in the country and we have plenty of these.