The Portugal U21 international spent part of last season on-loan with Porto and appears to not be in Arsenal manager Mikel Arteta's plans for the new season.
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The Athletic says Stuttgart have approached the Gunners to open talks about a fee for Vieira.
Arsenal will face accepting a loss on the €40m they paid Porto three years ago for Vieira.
The attacker made 26 appearances and scored four goals for Porto last term.