The WSJ union claims Burkle has told it he is interested in the possibility of teaming up with it in a joint offer for Dow Jones. Other billionaires have also been contacted by the union, the Independent Association of Publishers' Employees.
The news comes as Murdoch, chief executive of News Corporation, continues negotiations with the Bancroft family, who control the destiny of Dow Jones with their 64% voting stake in the company.
Steven Yount, IAPE president, said: "Our union remains hopeful that the Bancroft family will conclude that a sale of Dow Jones is not necessary. But if the Bancroft family is to be persuaded that a sale must take place, we believe that there are alternatives to Mr Murdoch."
Having initially rejected Murdoch's $5bn offer, the family last week agreed to consider to all offers for Dow Jones and to negotiate with Murdoch to find a way of securing the editorial independence of the WSJ under his control.
Yesterday, Murdoch and his son James had a face-to-face meeting with Bancroft family trustee Michael Elefante, Dow Jones officials, and three family members who are directors of Dow Jones: Christopher Bancroft, Leslie Hill and Elizabeth Steele.
The 30-strong family is divided over the desirability of selling Dow Jones to Murdoch.
Rivals to Murdoch may find it difficult to match the value of his $5bn offer, which is a 65% premium on the Dow Jones share price the day before his bid became public on May 1.