The sale is in response to heavy shareholder criticism of the group, which culminated in the resignation last week of chief executive Andrew Teare.
Mr Teare is said to have been a barrier to a radical breakup of Rank's collection of leisure interests which currently include Mecca bingo halls, Odeon cinemas and Butlins holidays .
His departure has opened the way for a sale of Deluxe, whose poor performance contributed to the recently announced 20 per cent drop in third quarter profits to pounds 111m.
Deluxe's profits were badly hit by the company's failure to cope with a gigantic order for 50 million video copies of the film Titanic.
Executives at Deluxe, which is managed from the US, are preparing a buyout bid which would value the video business at up to pounds 800m, though trade bids may also be invited.
At the same time, the company has received an informal approach from John Garrett, who is the former head of its leisure division.
Mr Garrett left in February after clashing with Mr Teare.
Mr Garrett telephoned Rank's chairman, Denys Henderson, at the weekend to propose a pounds 2bn break-up of the company, which would allow it to focus purely on its core leisure businesses.
Mr Garrett, who was one of the other candidates for the post of chief executive when Mr Teare was appointed in 1996, has still to raise the financial backing necessary to complete the deal.
He is also seeking to delete a clause in his severance package which stops him conducting any business deals with Rank before next April.
Shares in Rank, which initially jumped to 545p on Mr Teare's arrival in 1996, have lost nearly half of their value and the company has slipped out of the FTSE 100 index.
This morning the shares will open at 248.5p, valueing the group at pounds 1.92bn.
Profits at Butlins have fallen in spite of a pounds 140m investment aimed at modernising the holiday camp business.
Mecca, the group's casino wing, has been hit by increases in gaming tax, while profits at its Odeon cinema chain have also dropped.
Rank is already putting other divisions up for sale, including 16 nightclubs, the Claremont casino in London and its North American travel business, US Holidays.
Plans are also afoot to sell the group's 50 per cent stake in the Universal Studios theme park complex in Florida.