No player came close to the 15 goals the Austrian striker netted for the Rossoblu last season and Bologna are desperate to keep him despite a reported £7.6million bid from United.
De Vaio told Italia1: “It's not a question of money, he's priceless for what he means to us.
“We want to move forward with him because we are convinced that he can get better and help us grow. He is always the first in training to push the others, he has no market value.”
The club’s statements come after Arnautovic’s agent revealed his desire for the player to change clubs this summer - even if not everybody is happy with the bid at Old Trafford.
“It's true that there is an offer from a well-known club,” Danijel Arnautovic told Austrian outlet Kurier.
“I'm in contact with Bologna to see if that [offer] can be realised.”