Nugent, who has one England cap, has scored three times in 21 appearances for the Clarets this season.
His last goal was an equaliser in the 1-1 home draw against Bolton on Boxing Day.
The former Preston striker had made no secret of his desire to stay at Turf Moor, while new manager Brian Laws made the re-signing of the player one of his immediate aims after taking over.
Burnley have also denied reports that Tyrone Mears is in talks with Sunderland.
The defender had been linked with a move to the Stadium of Light but the Clarets have revealed that they have had no contact with Sunderland.
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