The firm currently has 106 stores across the country, and the closures would form part of a deal to renegotiate debts owed by the company to its landlords - which must be agreed by 75 per cent of its creditors.
The toy giant has around 3,200 workers across the UK, which means that a decision to close around 25, roughly a quarter, of its stores would affect at least several hundred jobs.
South Yorkshire has four Toys 'R' Us stores - one at Sheffield's Meadowhall Retail Park, one at the Crystal Peaks shopping mall in Sheffield, one at White Rose Way in Doncaster and another in Doncaster's Frenchgate shopping centre.
Sources said that Alvarez & Marsal, a specialist adviser on corporate insolvencies, would handle the CVA proposal.
All of the affected Toys 'R' Us shops will remain trading throughout the Christmas period and well into the new year, they added.
People close to the process said that Toys 'R' Us's larger out-of-town stores would be disproportionately affected by the closure plan, owing to their weak performance.
One said that the warehouse-style stores opened in the 1980s and 1990s were now too big and expensive to run profitably.
A CVA is a mechanism enabling companies to organise their funding and operations while enjoying protection from their creditors.
It was unclear whether Toys 'R' Us's landlords and other creditors would be prepared to back the proposal, further details of which will become clear next week.
In Britain, retailers including BHS, Focus DIY and JJB Sports have all used CVAs to exit loss-making stores, although all three companies ultimately succumbed to the fast-changing retail environment.
Toys 'R' Us's UK operations are understood to have been loss-making in seven of the last eight financial years.
Accounts for the year to January show turnover of £418m, on which it made an operating loss of £0.5m.
The effort to overhaul its UK estate follows the filing by Toys 'R' Us's American parent for Chapter 11 bankruptcy protection in September - a process which shields stricken companies from their creditors while they attempt to restructure their finances.
The proposal for the UK business of Toys 'R' Us has emerged on another bleak day for the British high street, with Royal Bank of Scotland and Thomas Cook announcing that they planned to close about 300 of their branches and stores.