It has joined the campaign to end period poverty and will make the items available from a newly-installed vending machine on the second floor on Tuesday.
Last month, the Scottish Government committed another £4 million to combat the issue following the rollout of free products at schools, colleges and universities.
Kathy Murdoch, centre manager at Buchanan Galleries, said: “The team here at Buchanan Galleries believe strongly that every woman and girl should have access to sanitary protection.
“We are therefore proud to do our bit in helping end period poverty in Scotland by offering free sanitary products to any shopper in need.”
Communities Secretary Aileen Campbell said: “I am proud that Scotland is taking world-leading action to ensure sanitary products are freely available for those who need them.
“I warmly welcome the announcement from Buchanan Galleries and I would encourage other organisations to take this small step to provide free sanitary products wherever possible.”