Reo-Coker is available on a free transfer as his contract at Aston Villa has expired.
Bruce is keen to bring in the 27-year-old as he tries to strengthen his midfield this summer.
The Black Cats have also been linked with the likes of David Vaughan, Seb Larsson, Spurs striker Peter Crouch and Sevilla ace Alvaro Negredo.
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Liverpool have unveiled their new home kit for next season, sticking with the hugely popular retro styling of the current shirt.
The reputation of the Premier League means it is the most popular and one of the most watched leagues in the world.