If the deal goes through, Disney will have beaten rivals Viacom and AOL Time Warner to the network, which broadcasts content to 81m cable TV subscribers.
News Corp will get a much-needed cash boost of about $1.6bn (£1.1bn), which will help strengthen its almost-completed takeover of US pay-TV group DirecTV. This in turn will strengthen its satellite holding company Sky Global Networks, parent of Sky.
News Corp's plans to spin off Sky Global were almost thrown into turmoil earlier this year when Haim Saban, the joint-venture partner in Fox Family, exercised an option that required News Corp to buy his 49.5% stake in the business.
After months of wrangling between Rupert Murdoch and Saban, News Corp decided to put the business up for sale. The two sides agreed a value for the business in June.
Disney will acquire a 76% holding in Fox Kids Europe, the lucrative children's programming network, but not the Fox Kids programming arm. Saban will sell his stake in Fox Kids programming to News Corp in a separate deal.