The report claims that United have been keeping close tabs on the Turkish international but that Inter are also preparing a bid.
Kaya worked alongside Inter boss Roberto Mancini during his time in charge of Galatasaray and established a close working relationship with the Italian boss.
United have been closely linked with Kaya in the past and were interested in signing him in 2014 according to the Mirror.
The 24-year-old played an important role in Galatasaray winning the league and cup double this season.
Kaya has vast domestic, Champions League and international experience. The young defender has established himself as one of Galatasaray’s key players and has three Süper Lig titles and two Turkish Cups to his name.
The Galatasaray star has also played in the Champions League for the past three consecutive seasons reaching the quarter-finals and second round.
Kaya still has four-years left on his contract and makes €1.5 million per-season and is valued in the region of €10 million by his club.
The Turkish international defender has also been linked with Juventus and Manchester United in recent weeks.