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.
"He also played beside Charlie Miller who was also impressed by his ability.
"There are work permit issues but Rangers hope to overcome them."