Sir David Beckham and Gary Neville were part of a consortium that took control of Salford in May.
Fleig said: “I couldn’t be more excited to join Salford City FC at such an important moment. This club is entering a new chapter with clear determination and huge untapped potential.
Neville added: “Gavin is exactly the kind of leader we’ve been looking for – strategic, passionate, and forward-thinking.
“His belief in the club’s potential is only matched by his determination to make things happen. We’ve got big goals at Salford City, and with Gavin at the helm, we’re confident we can deliver on them.”