The BBC has exchanged contracts with Axa Investment Managers for the 16-acre site.
The value of the sale has not been revealed but the BBC is currently looking to find £500 million in annual savings.
The campus includes seven stages and currently includes a workshop, office and post-production facilities.
John O’Driscoll, global co-head of real estate at AXA IM Alts, said: “With the signing of BBC Elstree Centre we will be acquiring one of the oldest working TV and film studios in the UK, with a history dating back almost 110 years.
“The structure of the transaction allows the BBC to continue its production on the site for at least another 25 years while providing us with the opportunity to invest in the site to create new world-class studios.
“The whole area has a long and illustrious history of producing some of the world’s most celebrated films and television series and, under our stewardship, we aim to continue that legacy.
“The TV and film studios sector remains one of our conviction asset classes, being underpinned by increasing demand from an ever-broader array of production houses, content creators and broadcasters.
“This is coupled with a very limited supply of high-quality modern film and production space, particularly in and around London where new construction is hindered by a lack of available land in the right locations.” The Standard