Stoke City chairman Peter Coates says the club are determined to keep hold of goalkeeper Asmir Begovic this summer.
Manchester City defender Aleksandar Kolarov wants to return to Italy to play in Serie A.
Rangers said Traynor, who left his role with the Daily Record on Sunday and has ceased working as a BBC Radio Scotland presenter, would oversee the development of all of the club's media platforms from early next year.
A statement read: "James joins Rangers following a distinguished journalistic career with The Herald, Express and Daily Record newspapers as well as being a presenter with the BBC.
"In his role at Ibrox, James will be supported by the Rangers Media editorial team, the press office and external media advisers, Media House."