The bags, which cost 6p each, are strong enough to carry shopping including heavy items such as milk and potatoes.
But they are made of natural materials, which mean they can be turned into compost along with food waste, and have been approved for use as waste food caddy liners, the Co-operative said.
They are cheaper per bag than buying a roll of compostable food waste bin liners, according to the Co-operative.
The bags have been trialled in several areas and are now being rolled out to around 400 Co-operative Food stores in 81 English local authority areas where householders are required to use compostable bags for food waste.
The Co-operative Food's environment manager Iain Ferguson said: "Every compostable carrier bag used is one less conventional plastic shopping bag in circulation.
"We believe they will have a significant impact upon the number of plastic bags which end up in landfill sites every year."
Mr Ferguson said the move made the Co-operative the only major food retailer to offer carrier bags with the "seedling logo" mark that means it has been certified as compostable.
"Our aim is to enable our customers to recycle more of the products they buy from us - from the leftover food and packaging, right through to the bag they use to carry their shopping home."
The move forms part of efforts by the Co-operative to cut numbers of single-use plastic carrier bags, which has seen its food stores use 64% less bags last year than in 2006. Around eight billion single-use plastic bags are handed out across the UK each year.
In Somerset, one of the areas where the bags have been tested out, Steve Read, managing director of the Somerset Waste Partnership, welcomed the initiative.
"It has allowed customers to take home their shopping and then use the same bags to line their food waste caddies. It saves them having to buy special liners, and has encouraged more people to support our food waste collections," he said.