The Light Blues had a £400,000 bid for the young defender turned down earlier this week.
The highly-rated 21 year-old has also been the subject of failed offers from Hull City and Peterborough, while there is also interest from Wigan Athletic, Stoke City and Derby County.
Oldham are aware of the growing interest and are trying to maximise the player’s value.
The Latics have offered Edmundson a new long-term contract on improved terms, but he has knocked that back.
The centre-half, who is keen on a move to Ibrox, wants to leave in this window but it all depends if a deal can be struck with Oldham.
Rangers are refusing to get involved in a bidding war and will wait for the dust to settle before they decide if they are going to increase their offer.
If they do they are unlikely to up their initial bid by much as they feel they have already made a fair offer.
Rangers have already agreed deals to sign Jordan Jones from Kilmarnock and Jake Hastie from Motherwell.
There is also a deal in place to sign Greg Stewart from Birmingham City, while the Light Blues still maintain an interest in Hearts’ John Souttar.
Kyle Lafferty and Wes Foderingham are among some of the Ibrox squad who could be moved on.