Correctional Services said that “matters are under control” at Johannesburg’s Sun City Prison on Wed.
The full extent of the Spain forward's injury is not yet known, with some British media reports claiming he could face up to six weeks out.
Costa, who arrived at Stamford Bridge in a £32-million (R540-million) move from Atletico Madrid, scored in Chelsea's league wins over Burnley and Leicester to start the season and any absence would be a big blow for Jose Mourinho's side. Costa suffered a hamstring injury with Atletico at the end of the previous season that forced him to limp out of the Champions League final against Real Madrid after nine minutes and hampered his World Cup campaign with Spain.
Meanwhile, Arsene Wenger confirmed Olivier Giroud is expected to be out until at least December with an ankle injury but the Arsenal manager stressed he was in no rush to buy another forward.
Giroud, 27, who was Arsenal's top scorer last season with 22 goals in all competitions, is expected to be out for up to four months following surgery on the ankle he injured during last Saturday's 2-2 draw at Everton in the Premier League.
"Olivier Giroud had surgery today," Wenger told Sky Sports after Arsenal reached the Champions League group stage for the 17th consecutive season with a 1-0 win over Besiktas in the second leg of their play-off on Wednesday. It went well but he will be out, to be at his top competitive level, until the end of December or beginning of January."