The brother and agent Marko Arnautovic is urging Bologna to let him leave.
The Austria striker is a target for Inter Milan and Roma.
"Last summer we received two very important proposals from the Premier League and out of respect for Bologna we did nothing," agent Danijel Arnautovic told Sky Italia.
“We received more offers at the start of June and Bologna promised that if there was interest from a top club, they would consider the sale so Marko could go.
“Now this offer has arrived and we would like Bologna to understand the importance of this opportunity for him. It would allow him to compete at the highest level and win titles."