Dutch newspaper Der Telegraaf claims Chelsea must qualify for the Champions League to land him.
But the deal has been agreed by all parties and Lampard will be delighted that his squad will be strengthened after the club failed to bring in any new blood last month.
At the end of the barren transfer window, Lampard said: “Some things are not in my control. My first thought as always is what happens on the training pitch. It’s clear that we wanted to bring players in.
“I certainly wanted to bring players in. I think I’ve made that pretty clear over the last few weeks and it hasn’t happened.”
Manchester United have also been said to be tracking Ziyech, but the Blues have moved quickly to snap him up.
The 26-year-old grabbed the attention of Europe’s top teams on Ajax’s journey to the Champions League semi-final last season and has again been in fine form this season, with six goals and 12 assists in the Eredivisie.
Ziyech also impressed in two group games against Chelsea in the Champions League earlier this season.