The Blues have already agreed to pay the Dutch champions £33m to secure Ziyech’s signature at the end of the season.
The 26-year-old Moroccan international will become Frank Lampard’s first signing since becoming Chelsea manager last summer.
Chelsea were under a two-window transfer ban when Lampard took over, and although that restriction was successfully overturned following an appeal, the Blues were unable to sign any players during the January transfer window.
"I am delighted and proud to have signed for such a huge club as Chelsea," the attacking midfielder told the club's official website.
"I am looking forward to next season and hope we can achieve great things together."
Ziyech has impressed for Ajax since joining the club from FC Twente in the summer of 2016.
He was particularly outstanding last season as Ajax won the Eredivisie title, KNVB cup and reached the semi-finals of the UEFA Champions League.
Ziyech has already faced and impressed against Chelsea this season in the UEFA Champions League, playing the full 90 minutes in a 1-0 defeat at the Johan Cruijff Arena in October before supplying three assists when the two clubs drew 4-4 in the reverse fixture at Stamford Bridge in December.
In 31 matches in all competitions for Ajax this season, the former Heerenveen man has scored nine goals and provided 12 assists in the Dutch top flight.
Ziyech has represented the Moroccan national teams 32 times, scoring 14 goals.
He was a member of the Atlas Lions’ squad that played at the 2018 FIFA World Cup in Russia.
