The BBC's Duncan Kennedy, in Washington, says Mr Murdoch wants to use the Wall Street Journal to take on newspapers like the UK's Financial Times.
Combined with non-family members said to support the bid, this could give News Corp effective control of the business.
His News Corporation already owns 100 other newspapers worldwide, as well as television stations and film studios.
The news led to a sharp rise in Dow Jones shares, which gained $5.82 to $57.38.
Adding the Wall Street Journal, our correspondent says, will give Mr Murdoch further access to the top table of influential publications and, he hopes, the profits to match.
But a spokesman for the Bancrofts said family members and representatives of trusts, which between them control 64% of Dow Jones' voting shares, had yet to make up their minds.
"Any suggestion that the process has been completed or that a particular level of support has been established is at this point premature," the spokesman said.
The bid seemed in peril on Monday when a deadline passed for members of the Bancroft family to decide on the offer.
News Corp reacted to this by saying it was "highly unlikely" to go ahead with the bid.
News Corp's board is meeting to decide whether to proceed with the bid and force Dow Jones shareholders to vote on the deal.
The bid has attracted criticism from figures within the media industry who believe it could threaten the Wall Street Journal's editorial independence, although News Corp has vowed to uphold the paper's independence.
Split views
Bancroft members are split over the merits of the deal.
The WSJ said that the position of Christopher Bancroft and a number of trusts was likely to be vital in determining the success of the bid.
It said discussions were hingeing on a proposal for the Dow Jones board to create a fund to cover the Bancrofts' legal and banking costs, thought to be about $30m.
News Corp first tabled its bid for Dow Jones on 1 May. It has since won the backing of Dow Jones' board, although one director subsequently resigned in protest.
A rival suitor is waiting in the wings, with Brad Greenspan, founder of the MySpace social network, saying he has got backing from five investors wanting to buy into the business.