Championship club Forest beat Arsenal 4-2 on Sunday with caretaker manager Gary Brazil in charge following the departure of Mark Warburton last month.
Karanka, a former Real Madrid and Athletic Bilbao player, was fired by Middlesbrough last season as the club headed towards relegation from the Premier League.
The 44-year-old had guided Middlesbrough to promotion from the Championship in 2016.
(Reporting by Simon Evans, editing by Pritha Sarkar)
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