A son of a factory worker and a nurse, Muntyan became interested acrobatics and competed successfully in Kiev's city-wide competition, winning accolades in his age category. His family eventually relocated to live near a professional soccer grounds in Kiev, where young Muntyan and his friends would hang out, acting as ball boys to the elders. While once juggling a ball, he was approached by a soldier, who asked him if he was interested in taking up football as a sport. Muntyan said yes and was taken to Mikhail Korsunskiy, who was a famous local children's coach at the time. He quickly recognised Muntyan's potential.
Muntyan joined the Dynamo Kiev team as a 15-year-old, when the main team coach was Victor Maslov. Despite weighting only 60 kg (9.5 stones) and short of 170 cm in height, he was encouraged to play and his skills were further enhanced by the training. When five of then current squad left to join 1966 Soviet football team for the World Cup, Dynamo Kyiv managed to win a double (championship and the cup) with Muntyan stepping in from the reserves as one of the main players.