Greg Tufnell, brother of former England cricketer Phil Tufnell, is leading Richess Group, a newly formed vehicle.
Other bidders, including Matalan founder John Hargreaves, are understood to have fallen out of the running.
Sources close to the process say that if no buyer is found by Friday, BHS is likely to be liquidated, putting the retailer’s 11,000 staff out of work.
Administrators for the department store chain had hoped to strike a deal by the start of this week, but the emergence of the last-minute bid has delayed the process.
The Telegraph reported that the Richess Group is thought to be backed by a wealthy Portuguese family.
Greg Tufnell was managing director of Mothercare from 1997 to 2000 and held a similar role at Burton, part of the Arcadia Group, for three years before that.
He is a director of House of Britannia, an investment firm that owns brands including glovemaker Cornella James, which counts the Queen among its customers.
Advisers questioned
Mr Tufnell is working with Nick De Scossa, a former investment banker and chief executive of Bristol rugby club.
The development comes as MPs on the Business and Work and Pensions select committees prepare on Wednesday to question advisers to Retail Acquisitions, which bought BHS last year for £1 from Sir Philip Green, as well as the BHS pension fund trustees.
At a hearing on Monday, the MPs were told by Goldman Sachs banker Anthony Gutman that he had told a key executive from Sir Philip’s company about the risks of selling BHS to Dominic Chappell, owner of Retail Acquisitions.
Mr Gutman said he had told Paul Budge, finance director of Arcadia, about Mr Chappell’s history of bankruptcy and lack of retail experience.