The authority wants to buy 42-46 Fargate and 26-28 High Street.
The Fargate store was occupied by New Look and is currently used by a fashion retailer.
A member of staff said they had been called by council official and told they had a year before being asked to leave.
The High Street building was last used by a building society. It has been earmarked as a hotel.
BUSINESS NEWS: The Floow acquired and applies for £4.8m loan
The council has asked the mayor’s office for a £6m grant to cover the purchases - and other schemes in the city centre.
The council last year bought the former Clinton Cards shop opposite Marks and Spencer and is converting it into an events hub.
A report by the Mayoral Combined Authority states: ‘Whilst initiatives such as Heart of the City 2 and the Future High Streets Fund have been in development since well before the start of the Covid pandemic, the impact of the past 18 months has brought these issues into sharper focus, requiring even more flexibility in what projects need to be delivered, how these are funded and what additional activity is required to cement the planned investment’.
BUSINESS NEWS: Airport applies for £20m loan
It adds that £2,410,740 is required for the acquisition, strip out and white-boxing’ of the two buildings and preliminary survey work to inform any offer/deal and all legal and planning fees associated with the acquisitions.
The authority received £18m from the Future High Streets Fund to improve Fargate and High Street.