The Atletico Madrid winger heaped praise on the Argentina forward, describing him as "out of this world" after his side played out a goalless draw with Barcelona on Saturday.
Messi made a return to action against Getafe on Wednesday after nearly two months out with a hamstring injury, coming on as a substitute to score twice in a 4-0 Copa del Rey win, before again starting on the bench against los Rojiblancos.
Turan, who exchanged shirts with the Blaugrana star after the game, was enthusiastic in praising Messi's quality and stressed that he felt that the world has never seen a better player.
"Lionel Messi is out of this world, he's an alien," the 26-year-old Turkey international told reporters. "For me, he is the best player in the history of football."
Messi has been shortlisted for the 2013 Ballon d'Or alongside Cristiano Ronaldo and Franck Ribery and is looking to win the award for a fifth consecutive time when the winner is announced in a ceremony in Zurich on Monday.
Atleti and Barcelona sit level on 50 points at the top of La Liga, six ahead of Real Madrid, which plays Espanyol on Sunday.