Turkish sources say the winger is being lined up by Fenerbahce to join their Dutch contingent in Istanbul.
Fener already count on Dutch pair Robin van Persie and on-loan Tottenham striker Vincent Janssen this season. Narsingh would leave Swansea on-loan for the second-half of the season.
Narsingh has been in and out of the Swans first team this season after his arrival from PSV Eindhoven almost a year ago.
The 27-year-old was the first signing of the Paul Clement era when he joined from PSV Eindhoven in a £4million deal in January 2017, although the Swans had been keen on securing his services for some time prior to the head coach's arrival.