The private equity firm co-founded by Luke Johnson, the entrepreneur and former chairman of PizzaExpress, has submitted a final-round bid to acquire part of Pizza Hut UK.
Mr Johnson's Risk Capital Partners faces competition from at least two other private-equity firms, including Rutland Partners and Leeds-based Endless, which also want to purchase Pizza Hut's restaurants in the UK.
The restaurant giant Yum Brands – the owner of Pizza Hut, as well as the KFC and Taco Bell chains globally – put the 340-UK restaurant business up for sale this year by hiring PwC to run the auction.
The accountancy company and the private equity firms declined to comment.
Yum wants to offload the casual dining business in the UK, as the group seeks to focus its capital expenditure on the fast-growing markets of Asia, Africa and Latin America.
Pizza Hut UK has struggled for much of the past decade, although it has cited an "encouraging" improvement in sales over the last 12 months, driven by investments in improving its menu and staff training.
Yum is now evaluating a number of final-round bids and is understood to be close to granting exclusivity to a preferred bidder to acquire Pizza Hut UK. Its owner is seeking a buyer for the restaurants to serve as a master franchisee, although its pizza delivery business of more than 320 outlets is not up for sale.
Yum will retain the rights to the Pizza Hut UK brand, but is thought to have offered suitors the concession of not paying it a royalty for up to three years.
A Pizza Hut spokesman said: "We have received significant interest in the Pizza Hut dine-in business from several leading companies. Yum has narrowed this list down, and we have been in advanced talks with a small number of prospective investors."
Mr Johnson organised the acquisition of PizzaExpress in 1992, floated it on the stock market and expanded the chain rapidly before he stepped down as chairman in 1999.
Pizza Hut made a pre-tax loss of £22.19m for the year to 28 November 2010.