United is widely reported to have made an approach to try and bring Arnautovic back to the Premier League, where he previously represented West Ham United and Stoke City.
Erik ten Hag is short of options to lead the line, with Cristiano Ronaldo's future uncertain and Anthony Martial injured, but luring Arnautovic from Bologna will not be an easy task.
Speaking ahead of the Coppa Italia clash against Cosenza, Di Vaio made it clear that the Serie A club is not willing to negotiate over a deal.
"It is not easy to play in the market phase, there is a lot of talk, our will is to retain Marko. He is at the centre of our project," he said.
"This interest makes us proud above all because it is an idea born by Sabatini [Walter, Bologna's former sporting director] and carried out by the club to realise a dream that was to bring him here after two three years in China.
"It was a bet won, he responded on the field and we want to move forward with him. Our will is not to sell, but to strengthen the team to make an important season."
Pressed over whether the club's stance would change if a significant offer was presented, Di Vaio suggested that would not be the case.
"It's not a question of money, it's priceless for what it represents for us," he explained.
"We want to move forward with him because we are convinced that he can do better and help us grow. He is always the first in training to spur others, he has no market value."