The company said it was consulting" with staff to close its shop in Giffnock, as well as another store in Cheltenham in England - two of its nine UK branches.
The shops employ around 150 staff between them.
The firm said: "A decision on the future of the stores will be made after the company has consulted with team members to discuss the proposal. In the event that the decision is taken to close the stores, we will work with team members to explore alternative employment opportunities.
"We remain committed to the UK market and will continue to look for ways to offer high-quality natural and organics to our customers."
The Giffnock store opened six years ago.
Leigh Sparks, Professor of retail at Stirling University, said: "It must cost them a lot to get products into the shop as it is a long way from anywhere and the company's others branches, which are mainly focused in southern England. For the business as a whole, the UK is a small market and the Giffnock branch is a bit of an outlier."
In August, the business, which was launched in the US in1980 and now employs 91,000 people worldwide, was bought by Amazon. It entered the UK market by acquiring seven Fresh & Wild stores in 2004.