The 48-year-old is understood to be a target of the Scottish FA to replace Gordon Strachan, and the board at Ibrox as Rangers hunt for a new manager.
Sunderland are attempting to gazump both those institutions by asking the Irish FA for permission to speak with their head coach.
The Wearside club have been without a manager since sacking Simon Grayson on Halloween.
It is now being reported that the Sunderland board waited until Northern Ireland’s World Cup qualification campaign was completed before making their move.
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O’Neill’s side exited at the play-off stage to Switzerland after losing 1-0 on aggregate.
Sunderland would have to pay more in compensation than either Rangers or the Scottish FA due to a sliding-scale release clause in the manager’s contract.