Preparations to get the stadium ready for the finals have been overshadowed by the deaths of three construction workers on the site over the past few months.
Two workers died when part of the stadium collapsed in November, before Fabio Hamilton da Cruz lost his life - after reportedly falling - last month.
Valcke, along with Brazil's Minister for Sport Aldo Rebelo and former striker Ronaldo - part of the Organising Committee's management board - met local authorities on Tuesday and toured the arena.
The secretary general confirmed that Arena de Sao Paulo will be ready for the opening match on June 12 between hosts Brazil and their Group A rivals Croatia.
"We received the latest updates on the progress of the work being carried out, and I can say that the Arena de Sao Paulo will be ready for the opening match," he said.
"The next key date now is May 17 or 18, which is when we will do the test event (a Brazilian championship match between Corinthians and Figueirense) and check the structure.
"This will give us three weeks to address anything that is not working.
"It's a race against time and with only 50 days to go before the opening ceremony, we don't have a single minute to lose."
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