The High Street chain's owner Arcadia went into administration in November, putting 13,000 jobs at risk.
News of the sale of the three-storey building has prompted an outpouring of emotion on social media, with shoppers recounting how important the flagship store is to them.
The store, which boasted a DJ booth, nail bar and food stalls, was a retail sensation when it opened in 1994.
The sale of 214 Oxford Street, managed by agents Savills and Eastdil, follows the failure of Sir Philip Green's retail empire to secure funding to pay its debts after sales slumped during the pandemic.
The Oxford Street building also houses Nike and Vans stores.
Arcadia said that although it was in administration, and so all its assets are to be sold, that did not mean the shops in the building would have to close.
People have been sharing their feelings about the London landmark, which was often used as a meeting point for friends and was a must-visit for fashion-loving tourists.
Arcadia, which also owns Miss Selfridge, Dorothy Perkins and Burton, had already closed other Topshop stores across the UK, citing the impact of the coronavirus pandemic.
Its brands were struggling before the pandemic, partly due to competition from online-only fashion retailers such as Asos, Boohoo and Pretty Little Thing.
Beyonce launched her Ivy Park collection at Topshop in 2016
The flagship store is currently closed, in line with the rules about non-essential retailers
The Oxford Street store pictured during Pride in 2018