The deal is likely to be presented to family members on Monday.
The family, which has some three dozen adult members spread across the US, has been deeply divided over whether to sell to Murdoch.
The Bancroft family has controlled the storied newspaper publisher for more than a century and views it as a public trust.
Murdoch has said concerns he would meddle with the Journal's coverage are unwarranted.
Dow Jones publishes The Wall Street Journal, the Barron's investor newspaper, the MarketWatch.com financial news Web site, and owns Dow Jones Newswires.
Under the terms of the agreement, News Corp would buy all outstanding shares of Dow Jones's common and Class B stock for $60 per share in cash, Dow Jones said in a statement.
Last week, Dow Jones negotiators met with supermarket billionaire Ron Burkle and internet entrepreneur Brad Greenspan to explore a possible counteroffer to Murdoch's bid, but no concrete offer has so far emerged.