While the original bags – currently priced at 40 pence – use the blue and yellow of IKEA's logo, Hay's version will be made from a white and forest green woven fabric, with forest green straps.
"We worked with colours that we feel will be long-lasting," said Mette during the unveiling.
The Frakta bag is among a range of other furniture and lighting products Hay has created in collaboration with IKEA. Set for release in 2017, the collection also includes a lightweight wooden table, desk lamp, bench seating and a green-coloured chair.
"In the same way as I feel with Hay, I feel that the accessories can be like the glue in this collection," Mette said. "A lot of the furniture will be grey and white and green."
IKEA's popular Frakta bags were originally designed by siblings Marianne and Knut Hagberg. They are made from synthetic material polypropylene, making them suitable for uses ranging from waste disposal to storage for clothes.
Dezeen teamed up with Ikea to live broadcast its annual Democratic Design Day event, during which a collaboration with British designer Tom Dixon was also announced.
Details of Dixon's project, to be launched in August 2017, are currently "top-secret", although the designer did reveal that it will feature an aluminium-framed couch he hopes will supersede the brand's Klippan sofa.
Hay and Dixon are the latest in a series of designer collaborations for Ikea. The company launched a collection by British designer Ilse Crawford in 2015 and is also working with Dutch designer Piet Hein Eek on limited-edition furniture set to launch in 2017.
IKEA's design manager Marcus Engman spoke exclusively to Dezeen about how the company is using design to overhaul its reputation for "doing cheap stuff" earlier this year.
Although this is the first time the two companies have collaborated, Mette previously said IKEA was helping brands like Hay enter the Chinese market.
"They're educating people in how to make a nice home," she said. "I believe they are helping us."