The Black Cats were relegated to the Championship at the weekend and the future of current boss David Moyes is now very unclear.
And, according to Spanish newspaper Sport, Karanka could be offered the job if Moyes does depart this summer.
Karanka was sacked by Middlesbrough in March after a ten-match winless run, leaving them perilously close to relegation.
They are still six points adrift of safety with three left to play and look set to follow Sunderland into the second tier.
Karanka has been suggested as a candidate for the job at Athletic Bilbao, one of his old clubs, but he will be considered by Sunderland.