By Fenwick, who founded Bloomberg Ventures in 2008, was previously chief executive of Bloomberg LP, taking over from the company founder Michael Bloomberg in December 2001.
Wall Street Journal reported news of the talks earlier on Friday.
The top job at Dow Jones has been vacant since last July when then-Publisher and Chief Executive Les Hinton resigned in the wake of the phone-hacking scandal at News Corp's UK newspaper unit, which had previously run.
Hinton told a UK parliamentary inquiry in 2009 that any problem with phone hacking at the company's papers was limited to one case. It was later revealed that thousands of ordinary people and celebrities had been the victims of the voice mail hacking.
Hinton, who worked with News Corp for 52 years, was perhaps Murdoch's closest associate.
Bloomberg and Dow Jones compete with Thomson Reuters.
(Reporting By Yinka Adegoke; Editing by Steve Orlofsky)
