In a fractious news conference Fifa secretary general Jerome Valke said the main venue in Johannesburg, Soccer City, was yet to be completed, and 700,000 tickets were still unsold.
He also expressed frustration at media speculation over the state of England’s proposed training camp.
“If the question is: ‘Could we host the World Cup tomorrow morning?’ the answer is no,” Mr Valke said.
“I can make a list of things that aren’t ready for the World Cup.”
Fifa has been forced to postpone announcing a list of the training camps for the 32 teams because at least two – England and Algeria – have not finalised their choices.
Fabio Capello declared himself to be satisfied with progress at the Royal Bafokeng sports campus during a visit on Monday.
But the FA has still not finalised the deal because of ongoing concerns about construction at the site.
Sky News visited the newly built complex near Rustenburg last week.
We found the hotel was finished and furnished, but work was still underway on the grounds and several of the six training pitches.
The on-site medical spa is also partially built. A spokesman for the hotel’s owners said all of the work was on schedule.
Mr Valke, who joined the England coach on his tour of the complex, said: “The rooms are there – there is a welcome desk where you can pay, a breakfast area where you can eat.”
He said an Irish company who are “experts in grass” are working on two of the training pitches.
Referring to preparations for the whole tournament, he pledged that South Africa would be ready “even if we have to stay awake all night, every day”.
Fifa has become used to hitting back at allegations that South Africa is not up to the job of hosting the World Cup.
But this is the first time that an official from football’s governing body has admitted in such stark terms that not everything is going to plan.
Last week Fifa was forced to slash the prices of some unsold VIP tickets in the hope of selling them to South Africans rather than overseas fans.
Out of a total of 3 million tickets, 2.3 million have been sold, raising fears of half empty stadiums at some of the lower profile games.
Fifa has cited the global recession as the main factor deterring fans from travelling to South Africa rather than concerns about crime.
The opening game is on June 11 at Soccer City – the showpiece stadium that is still under construction. South African organisers say it will be completed within weeks.