Updated from 1:01 p.m. EDT

Shares of Dow Jones ( DJ) surged 12% Tuesday as Rupert Murdoch appeared to be closing in on his quest to own the Wall Street Journal publisher.

The deal has garnered enough support from Dow Jones' controlling shareholders, the Bancroft family, to clinch the agreement, the Journal reported Tuesday. Cable-news channel CNBC reported that the $5 billion sale to Murdoch's News Corp. ( NWS) is a "done deal," with a definitive agreement likely to come by Tuesday evening.

Shares of Dow Jones recently were climbing $5.94 to $57.50. News Corp. has offered $60 a share for the company, a bid that represented a 67% premium over its stock price when the bid became public in early May.

The Bancrofts control Dow Jones through a web of trusts that own a separate class of shares endowed with 64% of the company's voting power. According to the Journal , family members owning at least 32% of Dow Jones overall votes will support the deal.

That would be enough to clear the way for a sale when combined with votes from public shareholders -- who are assumed to be in favor of a deal.

A Bancroft family spokesman cautioned that it's too early to make a determination on the family's position.

"The process of canvassing the Bancroft family members and trustees as to whether they wish to commit their respective shares to the proposed News Corporation transaction is still under way," the spokesman said in a statement. "Any suggestion that the process has been completed and/or that a particular level of support has been established is at this point premature."

Representatives for Dow Jones and News Corp. couldn't immediately be reached for comment.

News Corp. reportedly has a board meeting scheduled for 4 p.m. EDT, where the company will decide how to proceed. Though there appear to be enough votes to push the deal forward, the global media empire wants to receive a higher level of support from the Bancrofts in order to demonstrate to the public that it has the family's blessing. As an institution of American journalism, the Journal is widely viewed as a public trust with an important role to play in the democratic process.

On Monday, a News Corp. spokesman said the company would be "highly unlikely" to proceed with its bid with the family's level of support at the time, which was estimated to be around 28%.

Earlier Tuesday, the Bancrofts and Dow Jones were reportedly hammering out a pact for News Corp. to cover advisory fees for the families. These advisory fees would be at least $30 million, the Journal, reported. Such a payment would be aimed at appeasing some family members who are reportedly demanding a higher price for their super-voting shares.

News Corp.'s rich premium for Dow Jones comes in the midst of a crisis for the newspaper industry as it grapples with the rise of the Internet and a sluggish advertising environment. Despite the generous offer, the deal met a chorus of opposition to a sale due to concerns about News Corp.'s editorial standards and integrity.

A majority of Dow Jones' directors recently voted in favor of accepting News Corp.'s offer, but two Bancroft family members that holds board seats, Christopher Bancroft and Leslie Hill, reportedly abstained from voting in a move that amounts to a "no" vote.

Also, Dieter von Holtzbrinck, a Dow Jones director who is the heir of a German publishing empire, resigned his seat in protest of the deal, citing concerns about News Corp.'s editorial practices.