The People says the Old Trafford defender has refused a new deal at United, who turned down a move by Newcastle for the England star.
Goodison boss David Moyes is a keen admirer of the versatile player, who could leave for free in the summer. Roy Keane is a big fan of Brown, but attracting the star to fight a relegation battle at Sunderland may be beyond him.
United have made Brown a final contract offer.
Tottenham are also thought to be watching Brown's situation, with Juande Ramos desperate for experienced defenders.