NEW YORK, July 30 (Reuters) - Members of the family that controls Dow Jones & Co. Inc. have been asked to decide by the end of Monday whether they will support News Corp.'s $5 billion bid for the news organization, a source familiar with the situation said Sunday.

Rupert Murdoch's News Corp., owner of Fox News, the New York Post and a sprawling global media empire, made its $60 a share bid for Dow Jones, the publisher of the Wall Street Journal, on May 1.

The board of Dow Jones endorsed the offer two weeks ago, sending the deal to the Bancroft family for approval.

The family, which has controlled Dow Jones for more than a century and holds 64 percent of its voting shares, has since been trying to decide between its numerous members whether to support the bid.

Michael Elefante, a Dow Jones board member and chief trustee of the Bancroft family's shares, asked for family members to decide whether or not they support Murdoch's bid by the end of Monday, the source said.

Bancroft family members met in Boston a week ago to hear presentations by their advisers on the deal and to be given the opportunity to sign an agreement saying how they would vote.

The Wall Street Journal reported that as of late Sunday night, nearly 28 percent of the votes favored the deal, citing a person close to Dow Jones's board. Common shareholders representing 29 percent of the overall voting shares are expected to vote for the deal in overwhelming numbers, thereby ensuring a majority.

News Corp. needs 30 percent to assure a comfortable margin, the Journal reported, adding it was unclear whether News Corp. would proceed with less than that.

Dow Jones shares, which closed up 1.8 percent on Friday at $54.70 have been fluctuating as investors try to determine which way the decision will go.

The only competing proposal that still appears to be on the table is from Internet entrepreneur Brad Greenspan who proposed lending between $400 million and $600 million to some Bancrofts to buy out other family members at $60 a share.

News Corp. and Dow Jones were not immediately available for comment.