Among those present at Monday's meeting at a Hilton hotel in central Boston were Leslie Hill and Christopher Bancroft, both of whom are also board members of the New York-based company that publishes the Wall Street Journal (WSJ).
The Bancroft family owns 25 per cent of Dow Jones but controls 64 per cent of the shareholder vote through a special class of shares that have 10 votes each, compared to one vote for shares traded publicly.
On his way out of the meeting, Christopher Bancroft described the talks as "very productive". But he also said he has not altered his opposition to Murdoch's bid and said the family was considering a proposal from the entrepreneur Brad Greenspan, the founder of MySpace (now owned by Murdoch).
However Bancroft said that Greenspan's offer to help finance a buyout of some family members "can't top" Murdoch's bid of $60 per share.
Hill has also been trying to tempt other investors into making an offer, including Ron Burkle, a supermarket magnate.
The Bancroft family's lead trustee, Michael Elefante, said family members and trustees "now have all the information they need to make informed decisions about the News Corp. proposal", and a decision is expected later in the week.
That decision could dramatically alter the US media landscape.
The Independent Association of Publishers' Employees union, which represents many Dow Jones reporters, and some WSJ journalists have voiced fierce opposition to a deal with Murdoch, claiming he would use the Journal to promote his own commercial interests.
The journalists also believe a Dow Jones owned by News Corp would erode the Journal's proud tradition of editorial independence.
Murdoch has agreed to some concessions with regards to editorial independence and some Dow Jones shareholders have reportedly encouraged the family to sell out to News Corp.
As well as the WSJ, Dow Jones controls the Dow Jones newswires and a variety of other interests.
Murdoch has offered $60 per share for the company which is a large premium of about 65 per cent over the mid-$30s price that Dow Jones shares had seen prior to the offer becoming public.