The talks were reported by the Journal itself in an article posted on its website late last night.
The deadline set by the Bancroft family lawyer Michael B Elefante passed at 5pm yesterday (10pm UK time) with family members controlling only 29% of the voting shares in favour.
Based on calculations of the voting intentions of other Dow Jones shareholders, it is estimated that Bancroft family members holding at least 30% of voting shares would have to back Murdoch's offer.
A News Corporation spokesman said earlier yesterday it was "highly unlikely" Murdoch would proceed with the deal unless he received a greater level of support. This sent Dow Jones shares down 5.31% to $51.56, a long way below the $60 price News Corp is offering.
Subsequently a late-night proposal was being discussed in which the Dow Jones board would create a fund to cover payments to legal and financial firms advising Bancroft family members over the deal, estimated to total at least $30m, which News Corp would assume responsibility for if it bought Dow Jones, according to the Journal.
Dow Jones is expected to argue that the Bancrofts, which control about 64% of the voting shares, deserve help with their advisory fees because their trusts are so complicated and common shareholders don't face many of these fees.
But according to the report, the money would be paid only if at least one of two key shareholders agreed to the deal - Christopher Bancroft and a group of trusts managed by law firm Holme Roberts & Owen - because that would yield enough support from the Bancroft family to enable the sale to Murdoch.
As of early this morning neither key shareholder had agreed.
Separately, internet entrepreneur Brad Greenspan has renewed his interest in Dow Jones, saying yesterday he had gained the support of five investor groups interested in investing in the company to give it an alternative to Murdoch's $5bn (£2.5bn) offer.
The five possible investors are reported to be Intel Corp, Intel Capital, Apex Partners, Jana Partners, a unit of Softbank Corp and Trafelet.
News Corp first offered its $60-a-share bid back in April.
The family has been divided over the sale, with some members concerned about editorial independence at The Wall Street Journal if Murdoch was to assume control..jpg)
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