Owners of the store, in Harpur Street and Silver Street, are looking at a number of financial options and have appointed KPMG to help assess all options.
According to the Press Association and other national media, the Bournemouth-based group is looking to close a small number of its 22 stores and will also seek to slash rent.
Conversely Beales, was founded in 1881, has also said it wants to buy new sites, to expand a format that has seen early successes in its new Fareham branch.
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The group added it is looking for investment to "deliver a sustainable business model for the future" with a refocus on homewares, small domestic appliances, fashion accessories and shoes.
The Bedford store, which was previously Braggins, has been in the town for 37 years.
Mr Ezra Braggins bought part of the building in 1885 and opened the shop from 8am to 8pm during the week and 8am to 10pm on Saturdays.
In the early days it was a drapers and boys outfitters, but in 1914 the High Street shop selling girls clothes moved to the corner store.
After the Second World War, Braggins expanded in Silver Street when in 1947 Bryants Fish Shop was bought. In 1950 the Victory Hall was taken over and in 1955 new departments were opened for ladies clothes and fabrics.
In 1970, a new £85,000 extension in Harpur Street was built.