According to the Daily Mail, Hull City, Stoke City and Sunderland are all believed to be keen on Edmundson, who has made 79 appearances for the Latics despite his young age.
He was named in the League Two Team of the Year, despite Oldham experiencing their worst league season in nearly 50 years.
Fleetwood Town, managed by former Rangers midfielder Joey Barton, are also monitoring Edmundson’s situation and could look to pip the Ibrox side to the youngster’s signature.
The Light Blues will almost certainly see Gareth McAuley depart in the summer, while fellow centre-back Joe Worrall will return to parent club Nottingham Forest after the expiration of his loan deal.
Ross McCrorie is preferred in midfield by Steven Gerrard, leaving the Gers with just Connor Goldson and Nikola Katic as first-team central defenders.
Kyle Bradley is next in line, but the 20-year-old lacks senior experience having spent just one season out on loan at League Two side Annan Athletic. Aidan Wilson has made two appearances for the first-team but is also some way off regular first-team action, so it’s clear Rangers need to bring in reinforcements at the back.
Edmundson is currently valued at around £45,000 (according to Transfermarkt) but Oldham are likely to seek around ten times that amount for the highly-rated Manchester-born defender, who has 12 months left on his contract.
While he would be unlikely to unseat either Goldson or Katic as a first-choice centre-back, the number of games he has played already would be a good platform, and he would undoubtedly grow as a player at Ibrox.
Katic is one example of a young player brought in who has been nurtured and developed - he put in an assured performance in the 2-0 win over Celtic on Sunday - while the acquisition of 23-year-old Glen Kamara in January, and the performances of 21-year-old Ryan Kent and, to a lesser extent, 19-year-old Glenn Middleton have shown that there is a place for young players to shine at Ibrox.