The ex-Chelsea central defender is out of contract at the Premier League outfit Crystal Palace.
Gary, who won the UEFA Champions League with the London giants Chelsea back in 2012, is now a free agent, and the Gers want to secure the player’s services.
The Scottish Premiership giants could give the 35-year-old star £35,000-a-week as salary.
The Eagles want the former Bolton Wanderers ace to stay, but their contract proposal is less basic cash with extras for matches.
Gary is 35 now, and it is clear that the Englishman is past his best. However, he could still do a good job in the division, if his game time is managed properly.
Rangers could play in the UEFA Champions League next term and will face a big challenge from arch-rivals Celtic for the Scottish Premiership title.
Steven knows Cahill well, and the manager could make use of his big-game know-how and experience next term.
Cahill netted one goal and gave one assist in 20 PL appearances for Palace last season.
In other news, winger Kai Kennedy has agreed on a deal to sign for Dunfermline on loan until the end of the campaign, both clubs have confirmed on their official websites.
The attacker has joined the Pars and go straight into their roster for this weekend’s game versus Stenhousemuir in the Premier Sports Cup.
The Gers have high hopes for Kai have turned down bids to take the player away from Ibrox on a permanent basis, with West HamUnited the most interested in him.
Kai impressed during a loan stint at Inverness Caley Thistle in the first half of last term before an injury interrupted stint at Raith Rovers in the second half of the season.
But now he is focusing on impressing at the East End Park outfit and showing what he can do as he attempts to prove he has the ability to make the step up to the next level.