Only then will the public find out if the historic statues will be returned to the civic heart of the city or removed for good.
As well as Mr Matheson, the judging panel features David Mackay, partner of MBM Architects from Barcelona, Professor Andy McMillan, former head of the Mackintosh School of Architecture at Glasgow School of Art, David Harding, former head of environmental art at Glasgow School of Art, and Geoff Ellis, boss of DF Concerts.
Four of the firms on the shortlist come from the UK, one from France and one from America.
The six shortlisted designs have been on public display in the city's Lighthouse design centre for more than a week and have prompted a wide range of comments from the public.
Mr Matheson said: "The calibre of the six designs is very high and I'm sure whichever design team is chosen, Glaswegians will have a George Square to be proud of again.
"Each bidder has a wealth of international experience and all six have delivered a number of eye-catching civic spaces.
"This redevelopment is a hugely exciting moment in the growth of the city as Glasgow strives to forge ahead and meet its future challenges."
One of the first steps in revamping George Square will be to remove the statues for conservation work.
Only the cenotaph will remain and it will be kept under wraps to protect it during construction work.
The square will be out of bounds until next year when it will be unveiled in time for the Commonwealth Games.