In premarket trading following the report, shares of Time Warner jumped more than 12 percent. (Click here for the latest quote.) (TWX)
The sources said that 21st Century Fox offered $85 a share in cash and stock for the media giant.
In order to appease regulators on antitrust issues, 21st Century Fox said it would sell its CNN cable network, the sources said.
Chase Carey, 21st Century Fox's president and CEO, met with Time Warner CEO Jeff Bewkes in early June to discuss the deal.
Time Warner's board of directors sent a rejection letter in early July, the sources said.
Rupert Murdoch is "determined" to buy Time Warner, despite the board's rejection, sources said.
-Reported by Andrew Ross Sorkin; written by CNBC staff