Rupert Murdoch, the chief executive of News Corporation, will apologise for any wrongdoing committed at his News of the World newspaper, according to Reuters.
The news agency reported on Friday the apology would be published as a full-page spread in British national newspapers this weekend.
The apology over phone hacking follows the resignation on Friday of Rebekah Brooks as chief executive of the London-based News International, publisher of the News of the World before its closure last week over the scandal.
Another executive, Les Hinton, who was chairman of News International during the scandal, also quit as chief executive of the US-based News Corp's Dow Jones unit, The Wall Street Journal reported on Friday.
Hinton was head of News International from 1995 to 2007, when he joined Dow Jones, publisher of The Wall Street Journal.
Reuters said that under a large headline "We are sorry," the apology will contain the following message signed at the bottom by Murdoch himself.
"The News of the World was in the business of holding others to account. It failed when it came to itself.
"We are sorry for the serious wrongdoing that occurred. We are deeply sorry for the hurt suffered by the individuals affected," Murdoch will say.
"We regret not acting faster to sort things out. I realise that simply apologising is not enough. Our business was founded on the idea that a free and open press should be a positive force in society. We need to live up to this.
"In the coming days, as we take further concrete steps to resolve these issues and make amends for the damage they have caused, you will hear more from us. Sincerely, Rupert Murdoch."
Murdoch apologised to the family of murdered teenager Milly Dowler, one of many victims of phone hacking, but some had condemned him for moving to shut down the News of the World while Brooks kept her job.
Labour leader pleased
Brooks was the most senior executive at News International, a subsidiary of the US-based News Corp, which Murdoch heads, and is a former editor of both The Sun and the News of the World.
News International also publishes The Times and The Sunday Times.
Prior to the announcement of Brooks' resignation, Murdoch had vowed that she had his total support.
News of the World printed its final edition on Sunday , and some 200 hundred journalists were laid off as News International struggled to respond to the scandal.
But others within the Murdoch establishment had become increasingly critical of Brooks' ongoing leadership.
Alastair Campbell, communications director for ex-PM Tony Blair, discusses the scandal
Ed Miliband, the leader of the opposition Labour party, said he was pleased with Brooks' resignation.
"I am pleased that Rebekah Brooks has finally accepted responsibility for what happened on her watch as editor of the News of the World - the hacking of the phone of Milly Dowler, for example," he said.
"But, as I said when I called for her resignation 10 days ago, this isn't just about one individual; it's about the culture of an organisation.
"And when Rupert Murdoch says that News International have handled these allegations extremely well, I think people up and down the country will be thinking it really beggars belief."
Elisabeth Murdoch, Rupert Murdoch's daughter, called on Brooks to resign on Thursday, as did a major shareholder at News Corp, Prince Alwaleed bin Talal, the Saudi billionaire.
"For sure she has to go, you bet she has to go," Alwaleed told the BBC. His Kingdom Holding is the second biggest shareholder in News Corp and controls seven per cent of the votes.
In a statement, Brooks thanked Murdoch for his "kindness and incisive advice".
"As chief executive of the company, I feel a deep sense of responsibility for the people we have hurt and I want to reiterate how sorry I am for what we now know to have taken place," she said.
Brooks is due to answer questions in front of a UK parliamentary committee investigating phone hacking at the News of the World.
Murdoch also bowed to pressure to appear before the committee after David Cameron, the UK prime minister, said on Thursday he should attend and the committee said it would issue a summons compelling him to do so.
He told the Wall Street Journal, a News Corp title, that he had agreed to appear after being told he would be summoned, and in order to address "some of the things that have been said in parliament, some of which are total lies".
Murdoch's son, James Murdoch, News Corp's top executive in Europe, will also face questioning. The committee will begin hearings next week.
The about-turn by Murdoch sets the stage for a showdown with MPs who are keen to reduce the media tycoon's influence on British politics.
Murdoch, 80, has already been forced to back down on his plan to acquire BSkyB, the satellite television operator in which News Corporation has a 39 per cent stake, due to an outcry over allegations that reporters accessed private phone messages.
The hacking victims include celebrities, senior politicians, members of the British royal family and relatives of British servicemen killed in Iraq and Afghanistan.
Cameron has appointed a judge for a wide-ranging inquiry into the News of the World scandal and wider issues of media regulation.