Phillip Cocu only took the reins at the beginning of the season but after a woeful start his future at the club is already in doubt.
Fenerbahce has lost three on the bounce in the league and failed to make the Champions League group stage, losing to Benfica in the qualifying rounds.
Carvalhal is well regarded in Turkey, having managed Besiktas before moving to the English leagues, and would be considered a coup for the club despite the relegation at Swansea now being on his record.
However, according to 'Sky Sports', while the Portuguese would be flattered by the offer, he wants to continue his career in England.