The 21-year-old has been on loan at the Championship side since July but the player has now been offered a chance at Premiership level.
Livingston want to sign McCrorie with Rangers keen for the player to get top-flight experience.
"There is also a potential opportunity for Robbie McCrorie to play in the Premiership which would be brilliant for his development," he said.
"Robbie has done a great job at Queen of the South and they have been great for him but the opportunity to play a division above is great for his development and future potential at Rangers. Both McCrorie brothers are two model professionals of the game."
The goalkeeper is highly regarded at Ibrox and has a contract until 2022.
He has spent spells on loan at Berwick Rangers and Greenock Morton prior to his Queen of the South switch.
Game time in the Premiership with Livi, who had goalkeeper Matija Sarkic recalled by parent club Aston Villa, presents the next step in his development. However, he will face competition from another January signing Ryan Schofield, the goalkeeper joining from Huddersfield.
Rangers announced on Monday that goalkeeper Andy Firth has signed a new deal. The 23-year-old has been third-choice behind Allan McGregor and Wes Foderingham. The latter is likely to leave at the end of the season when his contract expires.