The teen signed with Chelsea last week and will move from Kairat when he turns 18.
Boranbaev told Sport-Express: "Interest from Zenith really was there, we talked. But the matter did not reach a specific proposal. I repeat, many paid attention to Dastan, but Chelsea turned out to be more serious than everyone.
"By the way, the first one of Zenit who drew attention to Dastan and advised the St. Petersburg club to acquire him was Andrey Arshavin.
"Yes. Arshavin immediately told Zenit: 'We need to buy this boy'. A year ago."
Asked about comparisons with Arshavin, Boranbaev also said: "It's hard for me to judge. Andrey — outstanding player. Dastan has a lot to prove. He has data. Dribbles with two legs, strong, prepared. I hope our pupil grows to the highest level. But it's too early to make bold predictions. The guy is 16 years old. He was already in Chelsea. Everything should work out."