It would mark the latest in a run of High Street buys for Next which has snapped up several chains including fashion chain Joules.
Last year, Next bought furniture brand Made.com, and a minority stake in baby goods store chain JoJo Maman B茅B茅 and more recently it increased its stake in fashion chain Reiss from 51% to 72%.
Next has been using the purchases to beef up its so-called "Total Platform", a suite of online services for third party brands.
This year it bought the floral fashion brand Cath Kidston - but not its stores.
Fat Face was taken over by a consortium of lenders in 2020 when shops were struggling during the Covid pandemic.
But in its latest annual results the retailer said sales were up 15% and profits had jumped.
Last year it was reported that Fat Face had hired investment bank Rothschild to find a buyer for the business.
Next, FatFace and Rothschild declined to comment on the deal.
UK high streets have been under pressure as shoppers spend less during the cost-of-living crisis.
Inflation - the rate at which prices rise - has started to ease after surging in 2022, but it remains high and retail sales are yet to bounce back.