The news pushed company shares up $7.69, more than 14 per cent, to $61 in pre-market trading on Friday.
The family said it remained committed to preserving the Journal's independence and integrity and were willing to meet with Murdoch to determine whether they could be assured those qualities would continue under his ownership.
They also said they would consider other offers.
Despite the Journal's prestige and tremendous clout in the business world, Dow Jones remains a relatively small company compared with large media operators such as Murdoch's News Corp.
"Dow Jones must remain an independent company if it is to prosper both as business and a journalistic enterprise"
Steven Yount, Wall Street Journal union leader
Reuters Group PLC and Bloomberg LP have made huge inroads in the business of providing real-time financial news and information to investors.
Murdoch started out as a newspaper owner in Australia, and today his News Corp media conglomerate has operations across the globe including Twentieth Century Fox, Fox News Channel, the New York Post, satellite TV broadcasting and the social networking website MySpace.
The $5bn offer that Murdoch made for Dow Jones could easily be paid out of News Corp's cash stockpile.
Murdoch has said he would invest in the Journal and ensure its editorial independence, something that the Bancroft family and employees of Dow Jones say is paramount to the company's mission.
The union representing Journal employees has been steadfastly opposed to Murdoch's overture, saying he would likely damage the paper's quality and compromise its independence.
Jim Ottaway Jr a former board member who controls 5 per cent of the company's voting power, has said he is also opposed to Murdoch's bid.
Like several other newspaper publishers including The New York Times Co and The Washington Post Co, Dow Jones is a public company but remains controlled by a family through a special class of shares with powerful voting rights.
Family control, its advocates say, is intended to insulate newspapers from short-term financial demands from shareholders and to safeguard their independence and protect their mission of public service.
Murdoch, for his part, has said he also understands the importance of family stewardship as his own company is controlled by him and family trusts, albeit to a lesser degree than the control exercised by the Bancrofts of Dow Jones, the New York Times' Sulzbergers and the Grahams of The Washington Post.
Also, Dow Jones has an important difference from the Times and The Washington Post in that no family members are currently involved with the company's day-to-day management. The Bancrofts are also numerous and geographically diverse, with some three dozen adult family members spread out over the country.
It remains far from certain that Murdoch will wind up winning Dow Jones, a company he has long been interested in owning.
In meeting with Murdoch and expressing a willingness to consider other bids, the Bancrofts may be trying to encourage other suitors to step forward.
|Dow Jones said it will consider the Murdoch bid|
as well as other potential alternatives [AFP]
Dow Jones said in a statement that its board would consider the Murdoch bid as well as other potential alternatives, and it would have a board representative present at the Bancroft family's discussions with News Corp and Murdoch.
A longtime newspaper owner with many papers in England and Australia, Murdoch has sought to allay the Bancrofts' concerns in a recent letter, saying he would take a series of steps to ensure the Journal's integrity, including setting up an independent oversight board.
Within Dow Jones, opposition to Murdoch remains strong.
Writers in the Journal's China bureau have expressed staunch opposition to his bid, and Ottaway and others say they're concerned he might soften coverage of China in particular, where Murdoch has business interests.