Edgar is available on a free transfer after leaving Turf Moor earlier this summer.
The Canada international helped Burnley win promotion to the Premier League last season, but is looking for a new club after his contract came to an end.
Edgar made 114 appearances for the Clarets after joining from Newcastle United in 2009, scoring four goals.
A number of clubs are thought to have expressed an interest in Edgar, but Birmingham are hoping to win the race for his signature.
Blues boss Lee Clark has been busy in the transfer market so far this summer after losing several players at the end of last season and he sees Edgar's versatility in defence as being a key asset.