The Brazilian has been impressive since arriving at the Turk Telekom Arena on loan, but their local rivals could steal a potential permanent transferhere.
Fenerbahce would reportedly be willing to pay Juventus' asking price for Felipe Melo, should Galatasaray not be able to afford the Brazilian.
The 28-year-old was a disappointment at the Turin club, but has undergone a renaissance since moving on loan to Istanbul, scoring four goals as Fatih Terim's men went into the winter break on top of the Super Lig table.
The Brazilian is set to spend the rest of the season at the Turk Telekom Arena, and has been widely expected to make the transfer permanent in the summer, but Galatasaray's local rivals Fenerbahce could now look to usurp the deal.
"Galatasaray are in negotiations to pay £10.9 million. Should they refuse to pay, we are prepared to pay this amount," a Fenerbahce source reportedly told Milliyet.
The Turkish champions' coach, Aykut Kocaman, also confirmed his admiration for Melo.
"Melo's ambition, fight, ability and style show a player with a full understanding of the game. Fenerbahce would love to see him," the former Turkey striker stated.
The Super Lig begins again on January 3, with Galatasaray entertaining Istanbul Buyuksehir Belediyespor. Fenerbahce face Orduspor the following day.