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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Since Guus Hiddink arrived at the helm, Chelsea have improved considerably after a tricky start to the campaign.
Newcastle United are staring down the barrel of relegation to the Championship for the second time in the past eight years.