The campaign to unseat members of the Murdoch family from their positions as directors of News Corporation may have spread to the US with cautionary advice about them from an influential adviser.

Glass Lewis, which advises institutions holding $15tn (£9.6tn) worth of investments, has recommended that investors vote against the re-election of Rupert Murdoch's two sons, James and Lachlan, at the News Corp annual meeting next week in protest at the phone-hacking scandal.

A report from Glass Lewis advises investors to "carefully consider the nature of the relationship each director has with the company and with its controlling shareholder, the Murdoch family, in order to establish a board with proper independence levels and strong oversight".

The move follows advice in Britain from the Local Authority Pension Fund Forum that Rupert and James should step down. The institutional shareholder advisory service, Pirc, has called for a shakeup.

The Australian Council of Superannuation Investors said last month that the News Corp board structure "does not reflect good corporate governance" and called for six directors to stand down.

Glass Lewis is also recommending shareholders carefully consider the re-election of Natalie Bancroft, David DeVoe, Andrew Knight and Arthur Suskind as well as the Murdoch sons on the grounds that they are not sufficiently independent.

Concerns about personnel on the board have existed for many years but the phone-hacking scandal unearthed by the Guardian and which led to the closure of the News of the World newspaper in Britain has been seen as a symptom of wider problems at the company.

A corporate governance specialist who asked not to be named said the move by Glass Lewis could trigger other influential organisations such as Institutional Shareholder Services (ISS) to come out against News Corp directors. The California Public Employees' Retirement System (Calpers), the largest US public pension fund and a leading campaigner on corporate governance issues, has yet to comment, but in summer it launched an attack on the dual-class structure that gives the Murdoch family almost 40% of the voting rights in the company despite owning only 12% of the equity. Calpers described this as "a corruption of the governance system".

News Corp declined to comment last night on the growing row before its 21 October annual general meeting but the Murdochs know their voting strength makes it difficult for investors to unseat the family members or other directors who have close ties with them.

The Murdochs have been under fire since July when it emerged that the News of the World had illegally targeted the missing schoolgirl Milly Dowler and her family in March 2002, interfering with police inquiries into her disappearance.

An inquiry was launched by Scotland Yard and soon led to the resignation of the then editor of the paper, Rebekah Brooks, who was more recently Rupert Murdoch's chief executive in the UK. The police investigation continues, and last week Lord Justice Leveson began a government-backed review of press behaviour and ethics.