The Colombia international made an impact at the World Cup, scoring three goals in four games before the penalty shoot-out defeat by England.
Barcelona only signed Mina in January from Palmeiras but could be forced to sell him as they have too many non-EU players in their squad.
And having paid €11.8m, Barcelona could net a near €20m profit if they decide to sell him.
Liverpool and Everton are both interested in the 23-year-old but Fenerbahce have made the first move, according to Spanish newspaper Sport.
This story has been reproduced from the media.