The 33-year-old forward is one of the best-paid players in the world on reported weekly wages of around 730,000 euros ($850,000).
The 33-year-old forward, one of the best-paid players in the world on reported weekly wages of around 730,000 euros ($850,000), has scored just twice since joining the Chinese Super League (CSL) club in January from Boca Juniors and has missed half of this season's games with injury.
His poor form and repeated absences have seen Shanghai fans turn on the Argentina international, raising questions over whether he would return after treatment.
Tevez has helped fuel the doubts after previously criticising the standard of Chinese football and hinting he might look for a move.
"No one ever said he would leave," a Shanghai spokesman told AFP.
"After this Saturday's game we will have a short break and the whole team will be off until August 28."
The CSL, which is more than halfway through the season with Shanghai in the bottom half of the table under Uruguayan coach Gus Poyet, pauses from this weekend until September 8.
The break, and Tevez's apparent injury problems, meant it was a good opportunity for him to seek rehabilitation back in Argentina and the decision to let him go was made after discussions between the player and the Shenhua's medical team, the club said.
Tevez, who has a history of falling out with clubs and coaches during an itinerant career that has included spells at Manchester City, Manchester United and Juventus, was due to be back in China by August 30, the spokesman said.
Nicknamed "Apache", Tevez fell out with his Manchester City boss Roberto Mancini when he refused to warm up as a substitute in the last moments of a Champions League match against Bayern Munich in September 2011.
Tevez flew back to Argentina and considered retiring, before eventually returning to England in February 2012.
He enraged Shanghai supporters in April when he was spotted at Disneyland the same day he missed an important match due to injury.
Poyet has also faced calls to be sacked or quit and Shenhua -- missing Tevez -- were smashed 5-0 at the weekend.
But they returned to action on Tuesday in the Chinese Cup and, with Tevez out of the country, beat second-tier Shanghai Shenxin 1-0 at home.