The sale would include the 20th Century Fox film studio and the Sky and Star satellite broadcasters in the UK, Europe and Asia.
Disney was left as the front runner after Comcast, the NBC owner, dropped out of the race on Monday.
The Financial Times said talks about the price were continuing on Tuesday.
CNBC reported that Fox and Disney were on a "glide path" for an announcement on Thursday, according to people familiar with the negotiations.
The Murdoch family was said to favour a deal with Disney because it would rather be paid in the entertainment giant's shares than Comcast stock.
A deal with Disney could also face fewer US regulatory hurdles, although it is extremely unlikely to be waved through.