Boro are seeking a new manager after accepting an offer from Wolves for Rob Edwards.
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The Sun's Alan Nixon is reporting Barry-Murphy is in the frame for the manager's job on Teesside. The former Manchester City academy coach has impressed with League One Cardiff, guiding them to fourth place.
Meanwhile, talkSPORT is reporting Valencia coach Carlos Corberan is also being considered by Boro.
Corberan left West Brom to take charge of Los Che last season and while he successfully guided them away from the drop, he has struggled to lift the team this season.