Up to 1,100 staff at Borders face losing their jobs on Christmas Eve unless a buyer is found this week.
The book chain went into administration on 26 November, before closing down sales were started three days later.
The firm’s administrators MRC said it was in ‘advanced talks’ with a number of possible purchasers about the sale of some of Borders’ 45 UK outlets, the BBC reports.
However, it added that any deals over the sale of shops had yet to be finalised.
MRC warned that unless store sales are agreed this week, all 45 Borders and Books Etc shops will close on 22 December, with staff finishing work on 24 December.
Borders went into administration after coming under severe pressure from internet competitors and supermarkets during the recession.
Waterstones owner HMV has repeatedly refused to comment on reports it is interested in buying some of the Borders shops.