The Spain international must now agree personal terms and pass a medical to complete the transfer.
The 24-year-old would be the Premier League champions' fourth signing of the summer, after goalkeeper Willy Caballero, defender Antonio Rudiger and midfielder Tiemoue Bakayoko.
Morata scored 20 goals last season after rejoining Real from Juventus.
He won La Liga and the Champions League, but the majority of his appearances were as a substitute.
Manchester United had been interested in Morata prior to signing Everton striker Romelu Lukaku for an initial £75m.
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