The popular firm already has four drive-thru takeaway locations south of the border, including in Newcastle, Greater Manchester, Derbyshire and Lancashire.
The first drive-thru north of the border is planned to be in Glasgow and to open from 5am until midnight every day of the week.
It would be built on top of a car park in the Great Western Retail Park in the West End of the city, beside a Burger King and Pizza Hut.
Glasgow City Council will consider the plans and a decision is expected to be made on May 29.