Manchester United have been keeping tabs on Kaya since he broke into the Galatasaray side back in 2011.
The Red Devils have reportedly set aside €15 million for the Turkish star and will look to finalise the transfer in January transfer window.
Atletico CEO Miguel Angel Gil was in Istanbul this week to discuss a possible stadium naming rights deal with Turkish Airways but was also rumoured to have held talks with Galatasaray over Kaya.
The former Galatasaray captain Arda Turan is also believed to be involved in negotiations.
The La Liga champions are willing to pay €10 million for the combative defender and will table an official bid in January.
The Turkish international defender still has four-years left on his contract and makes €1.5 million per-season.
The 23-year-old is into his third consecutive season in the Champions League and has 19 international caps.