The Scottish Football Association endorsed the application this afternoon, before being passed by the UK Border Agency.
But the 28-year-old still needs to obtain a visa in Australia before making the journey to Scotland.
Manager Ally McCoist had hoped the midfielder would be cleared in time to play in Sunday's Clydesdale Bank Premier League game against Motherwell.
However, it would now appear unlikely that McKay will be available for selection until next week's game against Aberdeen.
A statement on the club's official website read: "Rangers were successful today in gaining a work permit for Australian star Matt McKay but there is still some paperwork to be done before he can join up.
"The SFA endorsed Rangers' application and then it was passed by the UK Border Agency.
"However, the 28-year-old now needs to gain a visa in Australia before coming to Scotland.
"The Light Blues hope this process can be completed in the early part of next week and he can then travel to Scotland.
"Depending on when this takes place, McKay might be in a position to train with his new team-mates and stake a claim for inclusion in the squad for the Aberdeen game on August 28."
McKay secured his move to Rangers from Brisbane Roar last week after a fee was agreed by both clubs.