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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Five English clubs will compete in the last sixteen of the Champions League for the first time ever as Liverpool, Manchester United, Manchester City, Chelsea and Tottenham face some difficult away matches.