Zinchenko, who could be involved in Sunday's EFL Cup final against Arsenal, joined the club from Russian side Ufa in 2016 and has made his breakthrough in the City team under Pep Guardiola this season.
The 21-year-old told the Manchester Evening News that "living legend" Shevchenko inspired him to become a footballer.
"He was my idol as he was for everyone in Ukraine," Zinchenko said. "He has won the Ballon D'Or when he played for AC Milan. He is a living legend.
"First of all what he told me was that I had to be strong in my mind because here everyone is a superstar so you can not give up. You have to fight every single day, every single training session.
"This is the big difference between our football in the Ukraine or Russia. It's completely different. The English Premier League is the number one league in the world and don't forget Manchester City, in my opinion, are now the number one team in the world."
Shevchenko, who also spent time with Dynamo Kiev during his career, also appeared for the Ukraine national team in his time as a player.