Gers chiefs are seeking a replacement for Russell Martin, who was axed last week.
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Liverpool legend and former Gers manager Steven Gerrard was their favoured option, but he pulled out of the running on Saturday after meeting with officials in London the previous day.
Now former Manchester United midfielder and coach Carrick is emerging as a contender, with the Geordie available after leaving Boro last season.
Sky Sports says Carrick is now being discussed inside Rangers as an option for the Ibrox hotseat.