The veteran midfielder, 38, is set to be handed a new 12 month contract. Real have acted after Toni Kroos announced his retirement this week.
ANSA reports the Croatian has reached an agreement with Real Madrid president Florentino Perez also thanks to the approval of coach Carlo Ancelotti.
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Modric, who will turn 39 in September, could surpass Ferenc Puskas' record.
In 1966, when he was 39 years and one month old, the Hungarian became the oldest player to have played a match for Real Madrid.