Sports Minister Gayton McKenzie confirmed on Sunday that all players had now received the necessary visas required for travel through the United States.
The team had been scheduled to leave on a charter flight on Sunday for their training base in the Mexican city of Pachuca. However, the departure was postponed while the South African Football Association worked to resolve outstanding documentation issues.
In a post on social media, McKenzie said all players had received their visas, although several members of the support staff, including the assistant coach, team doctor, head of security and an analyst, were still awaiting final clearance.
Earlier, the minister criticised SAFA over the situation, describing the visa problems as embarrassing and unfair to both players and coaching staff.
The football association later confirmed that the team was unable to travel as planned due to visa challenges affecting several players and officials.
SAFA has yet to formally confirm the revised departure schedule, with an emergency meeting reportedly held to discuss the issue.
South Africa are set to begin their World Cup campaign in Mexico City on June 11 and will be making their first appearance at the tournament since hosting the event in 2010.
Head coach Hugo Broos had previously expressed a desire for the squad to arrive in Mexico by June 1 to allow sufficient time for recovery from travel fatigue and adjustment to the high-altitude conditions.
The team will now arrive a day later than planned. Pachuca sits at an elevation of 2,432 metres above sea level, around 680 metres higher than Johannesburg.
South Africa are also scheduled to face Jamaica national football team in a final warm-up match in Mexico on Friday before the start of the tournament.
The latest administrative setback follows an earlier qualification controversy, when South Africa nearly lost their place at the World Cup after fielding suspended midfielder Teboho Mokoena in a qualifier against Lesotho. The match result was subsequently overturned, but the team ultimately secured its place at the finals.
