Off contract this summer, Rudiger is in talks with Real about a new deal. However, there is also contact with Juve.
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Tuttosport reports the Italian giants are now firming up their interest in the 33-year-old ahead of this summer's transfer window.
Juventus has made initial contact with Rüdiger about a potential move to Turin.
They plan to offer Rüdiger a two-year contract, but it may be difficult for the parties to agree on the salary. It's suggested there is interest in the Premier League and Saudi Arabia.
Rudiger had a previous spell in Italy with AS Roma before moving to England to join Chelsea.