Sky News understands that Risk Capital Partners, the private equity firm founded by Mr Johnson, is on the brink of an agreement to sell Neilson to LDC, the investment arm of Lloyds Banking Group.
An announcement about a deal could come this week, although people close to the talks insisted that it could still fall apart.
The price that LDC is expected to pay for Neilson was unclear on Tuesday, although it was originally put up for sale with a price-tag of about £80m.
Risk Capital bought the activity holidays company, which sells skiing breaks and ten beach clubs across Europe, in 2013 from Thomas Cook, the then-struggling tour operator.
Under its existing owners, Neilson has substantially grown its profits, and now records roughly £80m in annual revenues.
Mr Johnson’s plan to sell the business comes amid a flurry of other mergers and takeovers in the UK travel sector, with brands such as Love Holidays and Away Resorts also being touted to prospective new owners.
Mr Johnson is more usually associated with companies in the restaurant sector, having built, invested in and transformed a string of well-known casual dining brands.
He made a big windfall from PizzaExpress, and has since owned chains including Strada and a stake in Gaucho, as well as holding outside roles such as the chairmanship of Channel Four.
The entrepreneur is now chairman of Patisserie Valerie’s listed parent company, which has been exploring a takeover of Gail’s, the artisan bakery chain also owned by Mr Johnson’s private equity vehicle.
The auction of Neilson has been handled by advisers at Cavendish Corporate Finance.