James Murdoch is being lined up for an expanded role at News Corp, the media empire controlled by his father, according to reports.
Mr Murdoch, 39, who gave up his main executive jobs in the UK earlier this year, is said to be taking charge of News Corp's US television businesses.
The Financial Times and News Corp-owned Wall Street Journal newspapers carried the reports.
On Thursday, Mr Murdoch was criticised strongly by the UK media regulator.
Ofcom said Mr Murdoch's record as head of the London-based News International newspaper group "repeatedly fell short" of what was expected and was at times "ill-judged".
If confirmed, the appointment would give Mr Murdoch responsibility for the successful Fox Networks Group, which includes cable channels such as National Geographic. It does not include the Fox News channel, which is a separate division.
The Financial Times reported that the appointment was likely to be confirmed later this year.
Mr Murdoch, deputy chief operating officer at News Corp, moved back to New York earlier this year after relinquishing his roles as chairman of BSkyB in April and executive chairman of News International in February.
Ofcom had been reviewing whether pay-TV company BSkyB, which is 39% owned by News Corp, was "fit and proper" to hold a broadcasting licence in the wake of the scandal.
The regulator allowed BSkyB to keep its licence, but criticised Mr Murdoch for failing to uncover the extent of phone hacking at the newspaper group.
News Corp welcomed Ofcom's decision on BSkyB's licence, but defended Mr Murdoch, saying some of the statements about him were "not at all substantiated by evidence".
The Wall Street Journal said News Corp had been waiting for "some clarity" on the fallout from the phone-hacking scandal before making a decision on Mr Murdoch's new role.