Ziyech spent four years in total at Ajax, making 165 appearances for the club and helping them win the Eredivisie title and reach the Champions League semi-finals.
The 27-year-old had a phone call with Lampard earlier in the year and says the conversation was key in convincing him to join Chelsea.
‘We spoke on the phone,’ Ziyech said. ‘The way he [Lampard] was speaking about the game impressed me.
‘For me also the most important thing was playing football. At that point he was just speaking about football and about his idea of the game. It was clear for me.’
Ziyech added: ‘I could not be any prouder [when he signed for Chelsea]. When you grow up, you always have a dream.
‘For every young footballer, you want to play in the Premier League or in big competitions. For me it was a really proud moment to sign for Chelsea.’
Ziyech made his Chelsea debut in last weekend’s 3-3 draw with Southampton, and also featured in Tuesday’s Champions League clash against Sevilla.
Discussing Ziyech’s Blues debut, Lampard said: ‘It’s hard, it’s hard because we are trying to find minutes for him.
‘He came on for his 20 minutes or so and think he showed his moments of class and calmness on the ball which you are going to see more of.
‘But it’s unfortunate for us he comes on and obviously unfortunate for us he comes on and we don’t win the game.’