In a New York Times interview, the News Corporation chief executive said he would propose to the Bancrofts, who own nearly 25% of the Dow Jones business and 62% of the company's voting stock, that The Wall Street Journal set up a separate board in order to protect the title's editorial integrity.
Murdoch said he would keep Richard Zannino, chief executive of Dow Jones, and newly appointed managing editor Marcus Brauchli in charge at The Wall Street Journal, as well as provide the title's journalists with "more resources and more investment".
Dow Jones' board of directors said earlier this week it would "take no action" in respect of News Corps' $5bn bid to acquire Dow Jones & Co, but according to reports, some members of the Bancroft family had yet to make a decision over the bid, prompting Murdoch to renew hopes of acquiring the business.
The News Corp offer, which was made on May 4, valued Dow Jones & Co at around $60 a share, but journalists at The Wall Street Journal have already written to the Bancroft family to express their concerns at the media firm's approach.
Murdoch has since moved to calm fears over a takeover, telling the paper, "We're not coming in with a bunch of cost-cutters, [but] I'm not saying it's going to be a holiday camp for everybody."
The Bancrofts have in the past rejected interest for the company, but it is believed only a few minds need to be changed in order for the acquisition process to get underway.
However, the major stumbling block for Murdoch's bid is likely to be opposition from Dow Jones & Co stakeholders, who are believed to be 58% against a takeover of the company.
According to sources, some members of the Bancroft family have rejected News Corps' initial bid in the hopes a rival bid may appear or that Murdoch's offer might increase. However, analysts have said this approach is "a risky proposition".
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