The Mirror says Morrison is out of contract at Premier League rivals West Brom after failing to agree new terms.
Morrison, who joined WBA from Middlesbrough in 2007 for £1.5million, is interesting former club Boro following their promotion.
But Black Cats boss Sam Allardyce is also a big fan of the ace, who has played consistently well in the top-flight for six seasons.
Versatile Morrison, 29, can play either central midfield or out wide.