Time running out for Sao Paulo's 2014 hopes
The Brazilians are also considering dividing the huge country into four regions, in which teams would be confined during the earlier rounds, to cut down on travelling.
Sao Paulo's plans were thrown into confusion last month when the committee and world governing body FIFA dropped the proposed venue, the Morumbi stadium, because the city had failed to provide financial guarantees for the renovation of the arena.
"If Sao Paulo wants to host the opening game or the competition as a whole, the deadline is getting close," Teixeira told reporters.
"Cape Town built its stadium in two-and-a-half years, so we are dangerously close to the limit."
Teixeira said he had a brief conversation with Sao Paulo mayor Gilberto Kassab last week but needed to meet state governor Alberto Goldman when he returns to Brazil before giving more concrete information.
"Nothing has been determined, it was a generic conversation because he (Kassab) is on holiday at the moment," said Teixeira.
"When I return to Brazil and he comes back from holiday, we will have a meeting with the governor of Sao Paulo."
FIFA general secretary Jerome Valcke said in a separate meeting that Brazil may be divided into four regions, rather than having teams travel around the entire country.
"Brazil is a continent not just a country, so we may divide it into four pieces, to make sure that fans do not have to travel (fly) more than one or two hours from one stadium to the other."
Teixeira, also president of the Brazilian Football Confederation (CBF), avoided directly commenting on a question about remarks by Valcke, who said in May that Brazil had fallen behind in its preparations.
"Next month, we will have the reports of the financial analysis of the stadium projects and we will see whether there is a problem or not," he said.
However, he admitted there were worries over the financing of the stadium in Curitiba.
Teixeira said the biggest problem facing Brazil was its ageing and often small airports, although he could not give any information on possible improvements.
"Airports are the responsibility of a government department called Infraero," he said.
He dismissed fears over Brazil's soaring crime rate, saying that violence happened everywhere.
"This problem exists all over the world. I see violence wherever I go, in Europe, in the United States, and in Brazil it's not different," he said.
Brazil was elected unopposed in 2007 to host the tournament which was earmarked for South America by FIFA under a short-lived rotation system. The same policy also brought the current World Cup to South Africa.
Brazil faces a huge job to improve its creaking stadiums and transport system while soaring urban crime is a major worry.