The Sunday Mirror says Campbell is set to join Sunderland for £7 million after the European Under-21 Championship in Sweden.
The Manchester United striker, who was sent off for two yellow card offences in Friday's semi-final win over the hosts, will be one of new boss Steve Bruce's key summer signings as he bids to add some pace and power to his squad.
Last week the Tigers seemed to move into pole position in the pursuit of Campbell. United boss Sir Alex Ferguson has accepted the youngster will not emerge as a first-team player at Old Trafford.
United offered Campbell to Wigan as a makeweight in their pursuit of winger Antonio Valencia.
However, United old boy Bruce believes he can push Sunderland into the lead of the Campbell chase.