Sky News has reported that Johnson, who stepped down as chairman of Gaucho in late 2017 after just over a year at the business, is among a small number of prospective buyers.
American buyout firm Carlyle and UK investor Aurelius Equity Opportunities have also reportedly made bids for the steakhouse group.
BigHospitality understands that a buyer is likely to be announced in the next two to three weeks.
Gaucho's previous owner Equistone acquired a majority stake for £100m in 2016.
Deloitte, which is handling the administration, said the 16 Gaucho-branded restaurants continue to “trade well”.
Johnson, who founded Strada and previously owned The Ivy, is currently partner at Risk Capital Partners, which backs Patisserie Valerie and Gail’s parent company Bread Holdings.
Speaking last month he said “nine out of ten” private equity groups had now “lost interest” in investing in the hospitality industry amid tough trading conditions.