O'Brien, an 11% shareholder in the company, first called for the sale of the company's loss-making title, along with a host of Australian radio stations, in June.
The Irish telecoms and media tycoon renewed his call yesterday and said that O'Reilly was too old to run a modern newspaper business and that, by leaving, he would save millions of euros, according to The Times.
He said: "The Independent has to go, as do other vanity projects. If he goes, and The Independent is sold, shareholders will save money. This is a company that is going nowhere."
Citigroup estimates that selling The Independent and its Sunday title would save £13.4m a year.
In June this year, O'Reilly threatened to sue O'Brien for publishing an allegedly defamatory report about IN&M, which raised concerns about governance issues within the company. It did not allege malpractice by the board nor the executives of the company.
O'Brien is the second largest share-holder after O'Reilly who owns about 28%, with other board members holding another 1.5%.
Independent News & Media operates in publishing, radio and outdoor advertising with its main interests in Australia, Ireland, New Zealand, South Africa and India, as well as the UK.