Campbell has had a major impact at the KC Stadium after spending much of the season on loan from Manchester United.
He has scored 15 goals for the Tigers and created the winner for Dean Windass in Saturday's play-off final victory over Bristol City at Wembley.
Former Halifax Town boss Brown is now planning to hold talks with his United counterpart Sir Alex Ferguson in an attempt to clinch a permanent deal for Campbell.
"I'll be doing business with United boss Sir Alex Ferguson," said Brown.
"I don't know if Fraizer will stay, that is for the manager of the European champions to decide.
"It is obvious that he is a talent and we want to keep him and I would have thought that we had the money to buy him now, but that is not my decision.
"He is top of my shopping list and I have promised the players who got us into the Premier League that they will be given their chance.
"We have a good nucleus, but I will add to the squad with players who will improve us on the field and off the field."
Meanwhile, Windass does not want to be referred to as a "legend" after his goal saw Hull promoted to the top flight for the first time in the club's 104-year history.
The 39-year-old Hull-born striker fired his hometown club to glory in his first-ever Wembley appearance, and milked the adulation of the 40,000-strong travelling support for all it was worth while he was out on the pitch.
The delight of the Tigers supporters was there for all to see that one of their own had shone on the biggest day in the club's history.
However, off the Wembley turf Windass, was quick to play down his achievement.
When asked what it felt like to gain legendary status among the Hull fans, he said: "I'm not a legend.
"People use that word too much in football and I don't like it. I'm not a legend, I'm a footballer.
"I don't like that phrase because you get people like those that go and fight for the country in Iraq and those sorts of places, and I get paid a lot of money to just kick a football about and do something I enjoy.
"They are the real legends - the real heroes. Not people like me, and I don't want to be called that."
Windass was substituted in the 71st minute and replaced by Caleb Folan. The veteran found it difficult to watch the end of the game as the promised land loomed large.
"I couldn't look. The last 10 minutes of the match were unbearable and I couldn't watch," he said.
"I was saying to myself, 'please blow your whistle ref, please, please'. When he did blow it, I just couldn't believe it."