The Spanish sports newspaper claims that the the Lions Istanbul rivals Fenerbahçe and Beşiktaş have also made a bid for the experienced keeper.
The report states that Valdes would prefer a move to the Yellow Canaries after being impressed by the ambition of the club.
Valdes is almost certain to leave United after falling out of favour with head coach Louis van Gaal.
Van Gaal was speaking during a press-conference where United’s new transfer Bastian Schweinsteiger was unveiled when he stated: “He [Valdes] is not selected because he don’t follow our philosophy. There is no place for that one player.
“Last year he refused to play with the second team.”
“When you are not willing to follow the principles of the philosophy there is only one way – out.”
Uruguayan international Fernando Muslera is currently the first choice keeper for Galatasaray and would be expected to keep his place in the team even if Valdes were to join.
The experienced keeper has been an instrumental part of the Lions recent success winning three league titles in the last four seasons.
The Istanbul giants will compete in the Champions League for the fourth consecutive year this season.