English League One side Portsmouth have had a bid accepted for the Oldham Athletic centre-back, report the The Scotsman's sister title Portsmouth News.
Pompey's offer of £600,000 plus add-ons has been approved.
Rangers were keen on the player and the club reportedly had two offers for the highly-rated 21-year-old rejected.
Edmundson attracted significant interest after his performances in League Two last season. He was named in the league's team of the year.
It is the second time Rangers are to lose out to an lowly English side for a player.
According to Gillingham, they beat the Ibrox side to the signing of midfielder Matty Willock following his release from Manchester United.