In a proxy proposal, influential investment group Christian Brothers Investment Services (CBIS) asked the media giant to appoint an independent chairman to lead the board.
Reuters notes that even though the nonbinding resolution is unlikely to secure the majority of votes, it could heap pressure on the board to remove Murdoch, who currently acts as chairman and chief executive of News Corp.
It also shows that News Corp shareholders remain concerned about the UK phone hacking scandal and the control Murdoch has over the business.
CBIS, which manages $4bn in funds mainly for Catholic institutional clients, is among a group of investors calling for major reform to the governance of News Corp.
Julie Tanner, who oversees socially responsible investing for Christian Brothers in New York, told Reuters that News Corp was "stepping into the scandal with a flawed corporate governance structure".
Christian Brothers has previously been able to use its weight to force change in governance at other firms, including convincing Goldman Sachs Group last year to make the board role of presiding director more powerful.
It has also pushed in many cases for companies to appoint independent chair roles to increase their accountability.
Murdoch has guided News Corp to become one of the planet's most powerful media empires, but the company has also been dogged by scandal, including the UK phone hacking scandal and recent allegations of pay-TV piracy at its NDS subsidiary.
A News Corp spokesman declined to comment on the proxy proposal.
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Separately, the News Corp director overseeing the internal probe into allegations of phone hacking and criminal behaviour at UK publisher News International has defended his role.
Viet Dinh told a panel in Washington DC that the fact he was godfather to one of Rupert Murdoch's children did not erode his independence.
"The key measures beyond the technical requirements of law is how you interact, how you act as a director, that is where true independence comes from," said Dinh, a former Justice Department official.
"There is no obligation that we hate each other to be serving on the same board."