The Bancrofts decided to meet Murdoch and his son James to discuss the possibility of a deal to sell the company, the talks are focused on concerns about the editorial integrity of the Wall Street Journal under Murdoch's ownership.
According to the paper itself, News Corp is willing to countenance an editorial board structure similar to the one used by Reuters, giving trustees veto power over any deal and ensuring they are consulted on the appointment and removal of the editor-in-chief.
In preparation for today's meeting, the family asked company executives to give them guidance on how best to preserve the newspaper's editorial independence were they to relinquish control.
Murdoch told the Wall Street Journal on Friday that he wanted the board to be made up of people with no business connections to him or to the Bancrofts.
Only three members of the 30-strong family, which remains divided over Murdoch's offer, will attend today's meeting. Christopher Bancroft, Leslie Hill and Elizabeth Steele are all directors of Dow Jones. The family's interests will also be represented by their trustee Michael Elefante.