Championship sides Rovers and Nottingham Forest would both like to sign the 31-year-old, with the former in the box seat to land the record Northern Ireland scorer on loan until the New Year.
Whoever Healy joins, Sunderland are expected to have to subsidise Healy's £20,000-a-week wages, which are beyond all but the top clubs in the Championship.
Healy was signed by Roy Keane for the Black Cats from Fulham in a deal worth around £1m in August 2008.
But the former Whites goal-poacher has never started a Premier League game for the Wearsiders with all 13 league appearances he has made coming from the bench.