Christopher Bancroft's plan has been reported by the Wall Street Journal, Dow Jones' main asset. The Bancrofts are demanding that the editorial independence of the Journal must be protected as a condition of selling Dow Jones to Murdoch's News Corporation, and this has proved a major sticking point in sale talks.
However, the Journal reported that Christopher Bancroft's idea is "a real long shot" according to Bruce Leadbetter, the partner of a private equity firm trying to help Christopher raise money, and to people "close to the Dow Jones board".
The shares owned by the family are "super-voting" shares, giving them 10 votes for every share, in contrast to common shares which offer one vote per share.
The "super-voting" shares lose this status if they are sold, unless they go to another family member.
If Christopher Bancroft succeeds in blocking Murdoch, common shareholders will lose the opportunity to sell their shares for the $60 per share price Murdoch is offering.
The Bancroft family controls 64% of the company's votes, so will have the ultimate say over Murdoch's final offer. Family members are split, with some happy to sell to Murdoch, some opposed and some preferring a sale to another party, though no serious competition to Murdoch's bid has yet emerged.
Another Bancoft family Leslie Hill was linked to another possible bid by Californian billionaire Ronald Burkle. However, he has said he will not bid alone.
Meanwhile, Murdoch is this week continuing discussions with the Dow Jones board as the due diligence process nears completion. He will meet Dow Jones chief executive Richard Zannino today to discuss price and other terms.