The supermarket will also introduce recycling points for customers for products not typically collected on bin days, including wrappers, empty crisp packets and face masks.
Bosses said if the trials in six stores in Edinburgh are successful, they will roll the format out to all 498 supermarkets across the UK over the next year.
The trials are in partnership with Nestle, which will be responsible for recycling soft plastics for the first time in the UK, with no waste shipped to other countries for processing.
Store waste will be sorted by staff in storage rooms, including soft and hard plastics, cardboard, food waste, green waste, PPE, tins, cans and foils.
To combat food waste, the supermarket said it will also offer more unsold goods through its Too Good To Go app, where surplus food is sold at a heavy discount.
Extra efforts will also be made to distribute the food to local communities, it added.
Jamie Winter, sustainability procurement director at Morrisons, said: “We believe that we can, at a stroke, enable these trial stores to move from recycling around 27% of their general waste to over 84%, and with a clear line of sight to 100%.”
He added: “All waste collected in our stores will be recycled here in the UK – we will not reprocess anything abroad. If we’re successful, we’ll roll this zero waste store concept out across the UK as fast as we can.”
Helen Bird, strategic technical manager at recycling charity Wrap, said resources for recycling plastic packaging and bags will be key to reducing waste going to landfill.
“Plastic bags and wrappers make up nearly a quarter of all plastic packaging that we use in our daily lives, yet only 6% is recycled,” she said.
“Until we have consistent and comprehensive household collections across the nations, supermarkets play a critical role to enable customers to recycle key items not collected at home.”
Morrisons said it has already reduced own-brand plastic packaging by 8,000 tonnes since 2017 and aims to cut operational food waste by 50% by 2030.
Bosses said they are also encouraging big-name brands to do more to reduce their own levels of packaging, and that Morrisons was the first supermarket to ditch plastic bags in favour of paper ones.