The Turkish midfielder helped the Rojiblancos to a shock La Liga triumph last season and a spot in the Champions League Final with his consistent displays.
Liverpool have been linked with his signature for some time, while speculation has suggested that Monaco are ready to move for him as they plot for life without James Rodriguez.
But now it has emerged that United are talking to the Spaniards regarding a deal for his services.
The Red Devils are understood to have spent the last 18 months tracking the versatile ace, and have been suitably impressed by his showings.
Atleti want at least £16m for the 27-year-old, which Monaco can fund easily given their vast Russian financial backing.
United are very keen, but their pursuit may hinge on selling Nani – who has been attracting interest from Arsenal.
Turan has previously spoken of a desire to play for Liverpool, which may not endear him to the Old Trafford faithful should a move go through.