The 28-year-old moved to La Liga from Stuttgart for an undisclosed fee in July 2010 but will soon be out of contract, allowing him to move for free.
Khedira scored eight goals in 146 appearances for Real and helped secure silverware during his five year spell at the Bernabeu, winning the Spanish League and Champions League, as well as the Copa del Rey on two occasions.
He has also enjoyed a successful international career, captaining the Germany U21 side to the European Championship crown in 2009.
Khedira was also a crucial part of the full Germany squad which won last year???s World Cup in Brazil, scoring in the 7-1 thrashing of the host nation in the semi-final.
He was forced to miss the final victory over Argentina though, after getting injured in the warm-up at the Maracana Stadium in Rio de Janeiro.
Khedira???s arrival in Turin could hasten the departure of Andrea Pirlo, who is expected to complete a move to MLS side New York City FC.