Manchester United have accepted an offer from the Tigers for Campbell, who is currently with the England squad at the European Under-21 Championship in Sweden.
The striker plans to wait until the tournament is over before deciding on his future after Newcastle and Sunderland were also linked with bids.
Brown is optimistic he can persuade Campbell to return to the KC Stadium after the youngster's impressive loan spell during the 2007/08 season.
Brown: "Sometimes a young lad's head can get turned by interest from other clubs and I'm convinced that other clubs will be interested in Fraizer.
"But the beauty of the whole deal is that I firmly believe Sir Alex Ferguson has the lad's best interests at heart with regards to his future.
"For him to recommend to Fraizer that Hull City is the best move for him speaks volumes for us.
"Hopefully he'll take a lot of notice of that advice."