O'Brien's aims have been perceived as pressuring IN&M, controlled by his rival Sir Anthony O'Reilly, to sell the loss-making Independent, according to a report in The Observer at the weekend.
The loss-making Independent could attract buyers if sold, although some media commentators including Peter Wilby, the former Independent on Sunday editor, and if O'Brien sells them.
However, The Observer report provides a new indication that O'Brien has set his sights higher. It quotes a source close to him as claiming he will increase his stake again soon, saying: "He isn't intending to sit on the sidelines with his tanks on their lawn for ever and a day."
O'Brien has taken his shareholding in IN&M to 22.15% since 2006, buying tranches of shares over the past few weeks.
In response, IN&M openly criticised O'Brien as a "dissident shareholder" who had made a series of "personalised, misleading and inaccurate" attacks on the company, and O'Reilly spent £15m late last week to boost his shareholding to 27.94%.
According to The Observer, this is enough to block a bid, but O'Brien could, by amassing more than 25% of IN&M shares, get the power to prevent it from making major transactions and disrupt its day-to-day running.