The 10-year-old and his family were forced to flee their homes because of the war with Russia.
City left-back Zinchenko, a Ukraine international, posted a picture on social media of himself playing football with the youngster.
Millions of people have fled Ukraine since the country was invaded by Russia in February.
Zinchenko wrote on Instagram: “This is Andriy. He is 10 years old. Today he is now safe. Like most Ukrainians, he and his family were forced to flee their homes because of the war.
“Seventy five days ago this boy dreamed of becoming a footballer and trained carefree with his team.
“Today he dreams of only one thing – peace in our country. About normal life at home.
“It hurts me a lot that because of the war there are a lot of children like Andriy in Ukraine today, deprived of childhood, and even worse, life.
“I believe that the day of our Ukrainian victory will come soon and all children will be able to relive their carefree childhood, full of children’s dreams and positive emotions, which today for a few minutes Andriy could feel at the training session.”
Zinchenko, 25, has been outspoken about the war in his homeland and been a leader for Ukrainians in the UK.
In March he also arranged for friend and former Shakhtar Donetsk team-mate Andrii Kravchuk to train with City’s Under-23s after leaving Russian club Torpedo Moscow.