Khedira, a member of Germany's 2014 Soccer World Cup-winning side, faces renewed competition for a first-team place at Madrid following the arrivals of compatriot Toni Kroos and Colombia star James Rodriguez and there have been rumours of an impending transfer in recent weeks.
He has been linked with a return to his homeland with Bayern Munich and has rejected an offer of a new Real contract, with his current deal set to expire in 2015.
"Part of me has already left Real," Bild quoted Khedira as saying during a visit to Leipzig to watch his brother play in a second-division match on Friday.
"Until August 31, anything is possible."
In choosing to travel to Germany on Friday, Khedira will no doubt cause considerable controversy among Real fans, with the La Liga team in action in the Spanish SuperCup at the same time, losing 2-1 on aggregate to Atletico Madrid.
"I gave him permission (to go)," Real coach Carlo Ancelotti said, adding however that he did not know where Khedira had gone.