The Dow Jones-owned newspaper said the $5bn (£2.5bn) bid would now go to Dow's board of directors for approval.
News Corp launched the surprise bid for Dow Jones in May.
Since then, Mr Murdoch's company has been battling to reassure sceptics that the paper's independence will be maintained under New Corp's ownership.
Family ties
Any deal would need the backing of the Bancroft family - which controls voting rights in the firm.
According to the newspaper report, the family will hold a meeting to discuss the latest offer on Thursday. The Bancroft's are expected to take several days to reach any decision on the matter.
Meanwhile, Dow Jones directors have been searching for a bidder to rival the News Corp offer, but no-one has come forward with an offer that trumps News Corp's $60-a-share price.
The offer represents a 65% premium on Dow Jones' share price at the time the deal was announced.
A successful bid would bring Dow Jones publications including the Wall Street Journal and Barron's under the same stable as News Corp's New York Post in the US, and The Sun and The Times in the UK.