The 20-year-old defender has been in excellent form since breaking into the Swans’ starting XI, and has caught the attention of a number of big clubs.
Despite him having penned a new contract in South Wales shortly before Christmas, the likes of City, Liverpool and Arsenal have not been deterred, and are all thought to be keeping an eye on the youngster.
But they may have to see off Atletico if they are to sign up the Welsh international.
The high-flying La Liga side have been watching Davies in recent times, and have been impressed by his fledgling displays.
Despite his young age, the left-back has already made 50 outings for Swansea since making his debut in 2012.
The Liberty Stadium outfit are far from keen to sell, but they know their hand may be forced if the player himself gets his head turned by the prospect of Champions League football.