The Bancrofts are reported to have discussed creating a system in which the board would have a public way to air disagreements with News Corp and have the power to hire and fire the managing editor of The Wall Street Journal.
News Corp submitted its unsolicited bid in early May, which valued Dow Jones at $5bn, or $60 a share, which represented a 67% premium.
The bid was rejected by the Bancroft family, which controls 64.2% of the voting shares, because some family members were concerned that a News Corp takeover would "crush quality and independence" at the journal.
Subsequently, the family met Murdoch earlier this month to discuss the offer and preliminary talks were said to be"constructive".
There has been speculation that several potential bidders have shown interest in the company, including: Brian Tierney, chief executive of Philadelphia Media Holdings, which owns the Philadelphia Inquirer and Philadelphia Daily News; and US retail billionaire Ron Burkle. However, no formal bids have been forthcoming as yet.
Separately, the Journal's newsroom is to receive a shake-up, with several top editors being reassigned or replaced, according to a report on The New York Times website.
The reorganisation is reported to be orchestrated by the managing editor, Marcus Brauchli, who took the job only a month ago.
The changes expected to be announced include the promotion of William Grueskin, the managing editor of the Journal online, to deputy managing editor, and naming Daniel Hertzberg to head the paper's Asia and Europe editions from Brussels. Two deputy managing editors are expected to lose their titles, according to the report.
Dow Jones stock fell below Murdoch's offer price yesterday for the first time in more than a week, losing $1.18 to $59.03.
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