The buildings sit at the north east corner of the square, between George Street and Martha Street.
Developer Chris Stewart Group wants to create apartments, a hotel and student accommodation, offices, and a pedestrian lane with cafes and bars.
It is now seeking planning permission from Glasgow City Council.
Chris Stewart, chief executive of the developer, said: "We have been looking for the right opportunity in Glasgow for some time now, and were delighted to be chosen as the preferred developer by City Property and Glasgow City Council for this important and strategic site right in the heart of the city centre.
"We have been working closely with Glasgow City Council and a number of other stakeholders for almost a year to ensure that our proposals and vision for the site match with the city's aspirations for high quality mixed use development."
Mr Stewart's firm wants to create a "George Street Complex" which would see two listed buildings renovated for five-star serviced apartments and commercial offices as well as the construction of a new hotel and student accommodation.
The plan includes a pedestrian lane with restaurants, bars and cafes and a central plaza.
It is thought that the development could eventually support about 320 jobs in the city centre.
Glasgow Chamber of Commerce chief executive Stuart Patrick is backing the scheme.
"These development plans offer an opportunity to regenerate a prominent city centre site, part of which has been left derelict for more than 80 years," he said.
"From a series of run-down buildings and an empty patch of land, the plans will create an area that thrives with people and businesses.
"The mix of offices, leisure and retail destined for this site, bringing both jobs and new visitors, will be of tremendous value to the city as a whole."
Planning proposals have been submitted for a £70m refurbishment of historic buildings at Glasgow's George Square which have lain empty for decades.