The Ibrox club has agreed terms with the 28-year-old who currently plays for Brisbane Roar.
And they will complete the signing once the international has undergone a medical and gets a work permit.
The Scottish Sun says Gers are happy to meet McKay's £320,000 buyout clause to make the deal happen.
He has been recommended to Ally McCoist by ex-Gers Craig Moore, Charlie Miller and Ian Ferguson.
Gers revealed they were after the player, who has 16 caps, and are desperate to sign him in time They said: "Rangers are hoping to make Australian international star Matt McKay their next signing as Ally McCoist continues to bolster the squad.
"The 28-year-old midfield star is firmly in McCoist's sights and a deal could be done this week. McKay is in the UK as Australia have a friendly international with Wales on Wednesday night and Rangers would love to do a deal before the Europa League play-off deadline on Thursday.
"McKay comes highly recommended by former team-mate Craig Moore who, of course, spent nearly ten years at Rangers, and Ian Ferguson who has watched McKay blossom.
"He also played beside Charlie Miller who was also impressed by his ability.
"There are work permit issues but Rangers hope to overcome them."