Weekly protests have continued across Brazil since June, the latest on Monday when police used rubber bullets, tear gas and shock grenades to break up protests in Rio and Sao Paulo. The protests were initially organized by striking teachers.
Valcke was met by 50 protesters on Monday when he toured a stadium in Cuiaba in west-central Brazil. Protesters scribbled protests slogans on freshly poured concrete just feet (meters) from where the pitch will be.
"World Cup. Why?" was one message.
Another read: "Less Cup, More Health and Education."
A banner paraded around the stadium read: "FIFA, Go Home."
Valcke defended the right of Brazilians to protest, even during the World Cup, but only if they are peaceful.
"What do you want us to do? What do you want me to say about it? It is happening. Will it happen at the World Cup? I hope not, but potentially it could happen," Valcke said. "There could be demonstrations during the World Cup."
He said FIFA had not requested any added security arrangements from Brazil for the World Cup.
Sports Minister Aldo Rebelo, sitting alongside Valcke, criticized reporters for focusing on the negative and for "almost rooting things don't go right."
Demonstrations "might happen," Rebelo said. "Yes, it's possible. But they might not happen also. ... Brazil, even having to deal with the problems of the demonstrations like what happened in June I think Brazil can have a World Cup in a peaceful environment."
Former World Cup winner Ronaldo, who has been living in London for several months, said non-Brazilians seemed more enthused about the World Cup than locals.
"What I have seen and heard abroad is a lot of enthusiasm from people," he said. "What I see abroad is very different from what I see here inside Brazil and what is being shown in the media. ... Of course the population wants other things like education, and security. But people also want to see the World Cup. This is a difficult moment with people going to the streets demanding their rights."
World Cup organizers said almost 6.2 million ticket requests had been made in the first phase of sales. Officials said 70 percent were from Brazil.
"There's a wish for the world to attend this World Cup," Valcke said.
Valcke reiterated that he expected six stadiums being readied for the World Cup to be handed over to FIFA by a Dec. 31 deadline. Six other stadiums for the World Cup were used for the Confederations Cup and are mostly ready to go.
"The question is not about being ready for the World Cup," Valcke said. "That's not the point. The stadiums will be ready for the World Cup."
Brazil is spending about $3.5 billion for new and refurbished stadiums. There are concerns that at least four of the stadiums could become white elephants. A judge in the insolated Brazilian state of Amazonas has suggested a stadium being built in the city of Manaus could be used afterward as a prisoner processing center.
Valcke said the stadiums needed to be delivered well before the World Cup opens on June 12, giving organizers time to test technology, install seats, ready lucrative hospitality areas and prepare television facilities.
He said a stadium being built in Curitiba in southern Brazil was under "permanent monitoring" by the organizing committee. He added that grass at the stadium in Brasilia had to be improved.
"The quality of the pitch in Brasilia is not what you are expecting," Valcke said. "We have more than enough time to make sure the pitches will be good."
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