Supermarkets with online sites are coming under extra pressure this year as more people try to order their Christmas groceries over the internet.
People are doing their Christmas shopping from home this year
The retail consultancy TNS predicts that Britons will spend £3.3bn next week - a rise of 17% from the same week last year.
But Sainsbury's has dismissed a press report which says online grocers are unable to cope with demand.
The grocer said it has responded by adding 30% to delivery schedules.
No room at the inn
The Financial Times insisted several supermarkets - Tesco, Asda and Ocado - as well as Sainsbury's - have already stopped taking Christmas orders following a massive rise in demand.
The paper quoted Tesco as saying its weekly internet orders had jumped from an average of 170,000 to 200,000 in the past two weeks, leaving its 1,500 delivery vans at full capacity.
Ocado, meanwhile, said it was delivering about 6,000 turkeys this Christmas, twice the number it dealt with last year.
Other retailers have also said they could do a lot more business if they had the physical capacity to deliver the goods ordered.
But the online supermarkets say they alerted regular customers to the potential problems in November and early December and encouraged people to book early.
This was a sensible move, TNS told the BBC, as cash-rich and time-poor customers do not want to be messed around.
However, some supermarkets admit they have already had to deal with customer complaints over the lack of delivery capacity.
"If there was a risk of failing to fill the Christmas orders that would be a PR disaster of heroic proportions," Ed Garner from TNS told BBC News.