The former Arsenal, Chelsea and Portsmouth man had been surplus to requirements at Madrid.
With one year left on his contract, Real opted to sell him and instead take Michael Essien on loan from Chelsea.
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The reputation of the Premier League means it is the most popular and one of the most watched leagues in the world.
Five English clubs will compete in the last sixteen of the Champions League for the first time ever as Liverpool, Manchester United, Manchester City, Chelsea and Tottenham face some difficult away matches.