Advisers to French bank Crdit Lyonnais will decide on a short-list of bidders this week after receiving over 30 offers from around the world.
Mr Branson's dream of repeating his Virgin Megastores success will depend on beating off tough competition from the likes of Michael Green's Carlton Communications and Sony, Japan's electronics and entertainment giant. Mr Green has secured the backing of Goldman Sachs and Polygram, the US entertainment group.
A Virgin spokesman said: "It's extremely exciting. We can bring in our retail skills and our brand name, as we did with the Megastores. We have already launched a number of music and video stores in a joint venture with the Gaumont cinema chain in France."
The MGM name reverts to the US parent company when the cinemas are sold. Rank is also bidding, but may run into regulatory problems from the Monopolies and Mergers Commission because it already owns the UK's second-biggest chain, Odeon.
The Virgin project was suggested to Mr Branson several weeks ago by Luke Johnson and Hugh Osmond, well-known in the City for their entrepreneurial flotations such as PizzaExpress and the printer Bath Press. They have recently been linked to a consortium seeking to buy Wembley stadium.
Mr Johnson and Mr Osmond have banker Hill Samuel and stockbroker Panmure Gordon lined up to arrange a listing for the MGM chain if successful.
Credit Lyonnais has appointed SG Warburg to sell the MGM chain, which has 416 screens on 120 sites in the UK. It also has another 30 sites in the Netherlands and Denmark. Sources said yesterday that different bidders were going for different bits of the chain. The Virgin bid is for the UK cinemas only and does not include the UK head office. Many of the UK sites are freehold, making them ripe for development.
A SG Warburg spokesman said yesterday: "There has been an incredible response from companies all over the world. We have had indicative offers and we are now putting together a shortlist. The sale is well on schedule for the summer."
It is understood most bids have been around £150m mark, and the highest about £160m.
Film industry commentators have suggested that the recent highly colourful history of the MGM cinema chain would make a jolly good film. The original owners ABC sold it to Thorn EMI, who then sold it on to Australian tycoon Alan Bond in 1986. Mr Bond's business empire bit the dust after a sustained assault from Tiny Rowland. Ownership then passed to Cannon, an independent company run by two Israeli businessmen Menahem Golan and Yoram Globus.
The Israelis ran into financial difficulties of their own at the end of the 1980s, and sold the MGM chain to an Italian businessman Giancarlo Paretti. Crdit Lyonnais backed him to buy MGM's Hollywood studios for $1.3bn (£867m), and when he went bust last year the bank seized the cinema chain in lieu of its loans.
Crdit Lyonnais would like a quick sale, since it is now at the centre of one of France's worst banking crises. City commentators have dubbed the Lyonnais saga a disaster "five times the size of Barings".