The 43,000-seater Arena da Baixada is so behind schedule that FIFA does not know when it might be ready and the organisation's secretary general said he will decide in three weeks whether to drop the city from the tournament.
"As you can imagine, the current situation of the stadium is not something we really appreciate," Jerome Valcke told reporters on a visit to the ground.
"The stadium is not only late, it is very, very late.
"If you don't have a stadium you can't have games," he added. "It is an emergency situation."
Brazil's Sports Minister said his government would do everything to avoid the potential embarrassment.
"All our efforts are focused on taking the measures that will guarantee the Parana stadium is in the 2014 World Cup," Aldo Rebelo said after meeting with President Dilma Rousseff and the head of the International Olympic Committee Thomas Bach.
The stadium is only 88.8 percent complete, according to owners Atletico Paranaense. It was supposed to be ready in December but like five other grounds due to be finished last month it is behind schedule.
One of the six, the Arena das Dunas in Natal, is due to be handed over to FIFA on Thursday.
The Corinthians Arena which is due to host the opening match in Sao Paulo will not be ready until April 15 after a fatal accident set construction back by months. The World Cup's opening game between Brazil and Croatia is scheduled to be held there on June 12.
The delay at the Arena da Baixada in the south of the country is all the more perplexing because the stadium is one of the few built in Brazil during the last two decades and is merely being refurbished for the 2014 tournament.
The venue is one of the least expensive of the 12 grounds at a cost of 326 million reais (£83.8 million). Four group games will take place there and world champions Spain are to be based in the city. Spain take on Australia there on June 23.
However, the reform has been plagued by delays. Valcke convinced Atletico in August to abandon plans to install a retractable roof because there was not enough time.
Judicial authorities ordered work to be halted in October because of alleged risk to construction workers. A month later, city councillors launched an investigation into allegations of over-charging and corruption.
Airports and public transportation are also a huge problem as Brazil prepares to host the tournament for the first time since 1950.
At least five host cities will not complete the promised bus lanes, tram systems or metro lines they promised.
Civil aviation authorities admitted on Monday that visitors to Fortaleza, where Brazil play their second match against Mexico, will arrive at an airport whose terminal building will be built from canvas because the proposed new structure will not be ready in time.
- Sports & Recreation
- Jerome Valcke