Asda says it has begun consulting with workers over a major restructuring of the business which could put about 5,000 jobs at risk, including 3,000 non-store jobs.
Proposed cuts would include in areas such as cash management, where there is less work as more people shop online.
The company said it planned to create 4,500 jobs as it expands its online offering.
The cuts could also affect two online-only stores where orders are picked.
Asda said its sites in Dartford and Heston that are used to pick online orders could close as the company switches to using regular stores to gather goods for online deliveries. This would mean the loss of 800 jobs.
It also plans to change about 1,100 store management roles, although it insists that this will mean an increase in jobs overall.
Asda chief executive Roger Burnley said: "The pandemic has accelerated change across the retail sector especially the shift towards grocery home shopping and our priority is to serve customers in the way they want to shop with us.
"The last 12 months have shown us that businesses have to be prepared to adapt quickly to change and I am incredibly proud of the way we demonstrated our agility and resilience through the pandemic."
People close to the business insist that the move is led by a change in demand from shoppers rather than a cost-cutting drive, and point to the 4,500 new jobs being created as order pickers and delivery drivers.
The company can handle 90% more online orders compared to a year ago.
Asda says it hopes many of those at risk can be moved to the new jobs and that the actual number of workers leaving the business will be lower.
One store manager, who agreed to speak on condition of anonymity, told the BBC the proposed job cuts would "pile more pressure and stress on an already stretched workforce" and described scenes of upset in their store.
Asda was recently purchased by billionaire brothers Zuber and Mohsin Issa and private equity firm TDR Capital, in a deal valuing it at £6.8bn, funded largely with debt.
The deal came after a failed attempt at merging with rival Sainsbury's.
"Asda workers have had a torrid two years," Roger Jenkins, national officer for the GMB union said.
"The failed Sainsbury's takeover, twelve months working on the pandemic frontline and now the uncertainty of a new take over, sidling the company with huge debts and potential sell offs.
"This is the last thing they need."
The union said the company is profitable and does not need to enforce redundancies.