Valcke, responsible for ensuring Brazil is ready in 11 weeks, met this week in Rio de Janeiro with organizing committee officials. Wrapping up the visit on Thursday, he assured that stadiums in Curitiba, Cuiaba and Sao Paulo would be ready when the World Cup opens on June 12 in Sao Paulo.
"We are late and we will have challenges," Valcke said. "And we will have a lot of work, and potentially some risks coming at the last minute because we have not tried and tested all the facilities."
Valcke, who two years ago said Brazilian officials needed a "kick in the backside" to speed up work, repeated that theme again, but in much softer tones. He also reminded that much of the work in Brazil had been slowed by disputes over who pays for what -- FIFA, local clubs or various levels of government.
"You cannot move the opening game of the World Cup to another stadium," he said. "It has to take place in Sao Paulo. There is no other choice. You have thousands of people who have bought tickets. ... We have to work all together to make sure it will happen."
One reporter asked about the possibility of FIFA-initiated lawsuits.
"Well, I mean we have enough legal action against FIFA and all of us in Brazil," Valcke said. "We don't need to initiate one ourselves."
Valcke said a fourth stadium, already opened in Porto Alegre, could be the most worrisome. He said Brazilian club Internacional and the local government settled a longstanding dispute about how to pay for temporary facilities, though no work has begun for the television compound, security, media and hospitality areas.
"All is written, all is signed and all the responsibilities or duties for each party are very well known," Valcke said.
He said FIFA hopes to avoid the confusion about who pays for what in the 2018 World Cup in Russia, which is also using 12 stadiums.
"It's a lesson and definitely we will act differently," Valcke said. "We will have to find a different way of working for Russia 2018."
Jose Maria Marin, president of the Brazilian Football Confederation, told Sao Paulo newspaper Folha de S.Paulo this week "we'll all go to hell" if Brazil fails to win the World Cup at home.
He offered the reminder again on Thursday.
"What (Brazilian) people really, really expect in the World Cup is to win the championship," Marin said, seated next to Valcke. "So we're still in purgatory. So either we all work together. ... or it's either to hell, or paradise or heaven. I'm confident we'll all go together to heaven."