Irishman Carey - who helped the Buddies win last season's Scottish Communities League Cup - travelled with the Tannadice outfit to Germany and Spain for the club's two-week summer warm-up camp.
However, McNamara has decided the versatile left-sided player - who can also play full-back - does not fit into his plans.
And 20-year-old French defender Jean Yves Koue has also been told he will not be offered a deal.
The Arabs issued the following statement on their official Twitter page: "Jackie McNamara confirms he isn't pursuing his interest in trialists Graham Carey and Jean Yves Koue at this time."
Former Turkish youth cap Nadir Ciftci, however, could be handed a deal after McNamara confirmed his interest in the former NAC Breda forward.
The 21-year-old remains on trial with the club while negotiations continue and has been joined by Chesterfield defender Neal Trotman in attempting to win a contract with McNamara's men.
The club statement added: "Jackie also confirms [the] club have defender Neal Trotman on trial this week and are hoping to keep Nadir Ciftci at the club if possible."