The team was due to leave on a charter flight for their training base in the Mexican city of Pachuca, but that has been delayed indefinitely while they resolve the issue.
"The South African senior men’s national team has experienced challenges regarding visas for some players and officials, and as a result the group could not travel to North America this morning as originally planned," the South African Football Association (SAFA) said in a statement on Sunday.
"SAFA is working around the clock to ensure that the team travels to Mexico City as soon as possible. We remain committed to ensuring that the team’s preparations for the tournament remain on track and in the meantime, Bafana Bafana will continue to train in Johannesburg until departure."
The organisation said they would hold an emergency committee meeting on Sunday night to discuss the issue.
“This SAFA travel and visa debacle is embarrassing and grossly unfair towards the players and coaching staff,” he said.
“I have informed SAFA that I need a report and action must be taken against those responsible for this mess. We are being made to look like fools.”
South Africa play in the World Cup opener against co-hosts Mexico in Mexico City on June 11, and are appearing at the global finals for the first time since they were hosts themselves in 2010.
An administrative bungle in qualifying almost cost them their place at the finals when their 2-0 victory over Lesotho was overturned after they fielded an ineligible player in Teboho Mokoena, who was suspended for the clash due to an accumulation of yellow cards.
Despite this, they still finished a point ahead of Nigeria and Benin in their qualification pool.
South Africa also have Group A games against Czech Republic (June 18) in Atlanta and South Korea (June 24) in Monterrey.