Chelsea reportedly pushed hard to bring the 26-year-old in during the January transfer window but they were unable to do so.
However they have now agreed a transfer for the summer the fee has been confirmed by Ajax to be €40m potentially rising to €44m depending on various performance bonuses.
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Chelsea confirmed the deal saying "The move is subject to the player agreeing personal terms and he will remain at the Dutch club for the rest of this season."
Ziyech rejected a move to Sevilla last summer, saying at the time that he would only leave Ajax for the right club.
The initial reports came on Wednesday from Dutch outlets De Telegraaf, VI and AD.
Ziyech becomes the first senior signing made by Frank Lampard since taking over as Chelsea boss.
The Blues had their transfer ban lifted for the January window but failed to make a signing.
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