Complete has appointed LongAcre Partners, the corporate finance advisory house to oversee the sale, which when it goes ahead could lead to a management buyout of Celador Productions and Celador International.
The Chris Tarrant-fronted 'Who Wants to Be a Millionaire?', which is now in its 19th UK season, has become a mainstay of Saturday night television. It regularly attracts an audience of more than 7m.
The programme, which was launched on ITV1 in September 1998, has since been licensed to 105 countries worldwide.
Paul Smith, Complete chairman, said: "I have been at the helm of Celador for over 20 years and it is no secret that I have, for some time, wished to reduce my responsibilities within the television division to allow me to focus my attention on opportunities within radio broadcasting and film production."
It is believed that 'Who Wants to Be a Millionaire?' earns Celador up to £25m in annual revenues. The company, which also owns the rights to Channel 4's 'You Are What You Eat', BBC One's 'Commercial Breakdown' and a new BBC Three series, 'Live! Girls!', is believed to have sought a buy-out for some time.
Smith added that he had already downscaled his day-to-day involvement with Celdor Productions and Celador International by devolving responsibility to the managing director of each company.
"Both are led by motivated and accomplished management executives, have healthy current projects as well as a number of exciting developments in the pipeline, and prosperous futures ahead of them," he said.
Celador has yet to announce when the sale is likely to take place.
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