The Brazilian is currently behind Nemanja Matic and Cesc Fabregas in the battle for the two central midfield spots at Stamford Bridge and could be tempted by the lure of regular first-team football in Turkey.
However, chances of him leaving West London appear to be extremely slim as he has entered talks with the Premier League champions over a new deal and that could result in Fenerbahce turning their attentions to his team-mate John Obi Mikel.
The Nigeria international is likely to leave the Blues during the summer after making just six league starts last season and with rumours that manager Jose Mourinho wants to bring in another central midfielder [via Telegraph Sport], Mikel’s first-team opportunities will be limited should he stay and a move to Turkey could re-ignite his career.
Fenerbahce have certainly been flexing their muscles in the transfer market this summer as they have already completed a deal for former Manchester United winger Nani [via BBC Sport] and are reportedly closing in on the signing of Robin van Persie [via Guardian Sport].
The Super Lig outfit are clearly signalling their intent to challenge Galatasaray for the title next season and the capture of two more Premier League players would increase their chances.
Ramires has been a key player for Chelsea since joining the club from Benfica, showing himself to be a fine box-to-box midfield player, though his all-action displays have not been enough to keep him as a regular starter in the team in recent times.