KSL Capital Partners is one of several private equity firms considering making an offer for the chain of fitness clubs, the Sunday Times reported.
KSL bought the Belfry in Warwickshire in early 2012, and last month snapped up the Malmaison and Hotel du Vin boutique hotel brands.
Hatfield-based David Lloyd Leisure, which has 80 clubs across the UK, is controlled by property group London & Regional and private equity firm Caird Capital.
Blackstone, a US investment house, is also said to be mulling a bid.
The fitness chain is named after former English tennis professional David Lloyd, who founded the business in 1980.
David Lloyd has a focus on racquet sports, with around 700 tennis courts, 180 badminton courts and 140 squash courts across its UK sites. It has about 440,000 members in the UK and Europe and employs 6,000 staff.
Despite the prolonged economic downturn, which has squeezed gym chains, the report said David Lloyd is "trading well", increasing its appeal to potential bidders. Rival Fitness First last year avoided administration with a debt restructuring which involved selling off 67 gyms and slashing rents.
Earlier this year it launched a joint venture with Talwalkars Better Value Fitness - India's largest health club operator - to develop leisure and sports clubs targeting the country's increasingly affluent population in cities such as Delhi and Calcutta.