Boss Arsene Wenger managed to hold onto the gifted playmaker in the summer despite the best attempts of Barcelona to capture Fabregas, who began his career as a youngster at Camp Nou.
Real boss Jose Mourinho is keen to strengthen in January but the former Chelsea boss has been left frustrated as the club do not intend to hand him a massive war-chest in the New Year.
Madrid based publication Marca claims Mourinho will be backed in the summer and they have suggested he will try and sign the Gunners captain by launching a £35million bid.
However, Barcelona-based paper El Mundo Deportivo have also claimed the Catalan giants will reignite their long-standing interest in Fabregas in the summer and Wenger could be set for another lengthy battle if he is to convince the 23-year-old to stick at Arsenal, especially if the Gunners fail to secure any major silverware this season.
If Real or Barcelona do launch a bid for Fabregas they will probably have to fork out a bigger fee than the £35million that has been reported, as Fabregas is on a long-term contract at the Emirates and the Gunners rejected Barca's summer bid in the region of £29million as they felt the Spanish international was worth considerably more.