The Argentine striker, who was scheduled to resume training tomorrow, will now return on Monday. The delay will cause him to miss out on the Community Shield fixture against Manchester United on Sunday.
A City spokesman said: "Carlos has now been given extended leave so he can return on Monday, which is 21 days after Argentina's exit from the Copa America."
The forward has repeatedly spoken of his wish to leave the Blues and move back to South America to be nearer to his two daughters.
A switch to Brazilian side Corinthians was close to being finalised before a dispute between the two clubs over the payment structure caused the deal to collapse. The South American side have since been unable to sign Tevez due to their transfer window closing.
Inter Milan and Real Madrid have also been linked to the former Manchester United striker's signature.