Foreign Secretary William Hague has defended David Cameron, saying he was "not embarrassed" by the extent of the PM's dealings with News International.
Mr Cameron has met its top executives 26 times in the 15 months since he became prime minister, it emerged.
Mr Hague defended the PM's decision to entertain Andy Coulson after the latter quit as an aide over the News of the World phone-hacking scandal.
In press ads, Rupert Murdoch apologises for "serious wrongdoing" by the paper.
The Mail on Sunday said on Saturday that its former news editor Sebastian Hamilton and ex-investigations editor Dennis Rice had been told by Scotland Yard that their phones may have been hacked by private investigator Glenn Mulcaire.
Mulcaire was jailed in 2007, along with former New of the World royal editor Clive Goodman, after admitting intercepting voicemail messages on royal aides' phones.
A list of engagements released by Downing Street shows that Rebekah Brooks, who quit as News International chief executive on Friday, had been entertained at the prime minister's official residence Chequers in June and August last year.
News International chairman James Murdoch also attended Chequers in November.
There were further social meetings between Mr Cameron, and James Murdoch and Mrs Brooks, last December.
Mr Coulson stayed at Chequers in March this year, two months after he quit as Downing Street director of communications following fresh allegations of phone hacking under his editorship at the News of the World.
"In inviting Andy Coulson back, the prime minister has invited someone back to thank him for his work - he's worked for him for several years - that is a normal, human thing to do," Mr Hague told BBC Radio 4's Today programme. "I think it shows a positive side to his character."
Mr Coulson was arrested last week as part of the police inquiry into phone hacking.
Regarding Mr Cameron's meetings with various News International executives, Mr Hague said: "I don't think that would be very different from previous prime ministers.
"Personally I'm not embarrassed by it in any way - but there is something wrong here in this country and it must be put right. It's been acknowledged by the prime minister and I think that's the right attitude to take."
The 26 meetings or events involving News International figures compares with: nine involving Telegraph Media Group figures; four meetings involving Associated Newspapers, publisher of the Daily Mail and Mail on Sunday; four meetings involving the Evening Standard.
BBC Radio 4 chief political correspondent Norman Smith said: "It's pointed out that News International is a formidable player in the British media market and therefore it's perhaps understandable that the prime minister should devote so much time to them.
"Similarly, when you look at the list, many of the meetings were for things like charity receptions or award ceremonies. They were sort of informal gatherings rather than serious, across-the-table discussions with senior executives at News International.
"Nevertheless, it sort of fuels the perception - certainly the accusation from Labour - that the prime minister was too close to News International."
The prime minister's relationship with Mr Coulson was "the one thing that could profoundly damage Mr Cameron from all these hacking allegations," our correspondent added.
"Still Mr Cameron seems unwilling to disown Andy Coulson, repeatedly stressing the good work he did in Downing Street and in no sense cutting him loose."
'Lack of judgement'
Labour's Lord Prescott accused Mr Cameron of being "very much wrapped into the Murdoch operations".
Shadow culture secretary Ivan Lewis said the disclosure of the list of engagements offered "yet more evidence of an extraordinary lack of judgement by David Cameron".
"He hosted Andy Coulson at Chequers after, in the prime minister's own words, Mr Coulson's second chance hadn't worked out.
"David Cameron may think that this is a good day to bury bad news but he now has an increasing number of serious questions to answer."
Rupert Murdoch has taken out full-page advertisements in several newspapers on Saturday, using the space to say: "We are sorry for the serious wrongdoing that occurred."
The printed apology expresses regret for 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," says the statement, signed "sincerely, Rupert Murdoch".
Rupert and James Murdoch and Mrs Brooks are due to appear in front of the Commons media select committee on Tuesday to answer MPs' questions on the hacking scandal.
Mrs Brooks was editor of News of the World between 2000 and 2003, during which time the phone belonging to murdered schoolgirl Milly Dowler was tampered with.
As well as her resignation, senior News Corporation executive Les Hinton quit on Friday.
Mrs Brooks has been replaced by Tom Mockridge, who was in charge of News Corporation's Italian broadcasting arm.
In a resignation statement, Mrs Brooks said she felt a "deep responsibility for the people we have hurt".
Mr Cameron said through a spokesman that her resignation was "the right decision".
Mr Hinton, chief executive of the media group's Dow Jones, said in a statement that he was "ignorant of what apparently happened" but felt it was proper to resign.
The most senior executive to leave the conglomerate, Mr Hinton was previously head of News International from 1995 to 2007 and has worked with Rupert Murdoch for more than five decades.
On Friday, Rupert Murdoch apologised to Milly Dowler's family at a meeting in London.
The family's solicitor Mark Lewis said the newspaper boss looked very shaken up and upset during the talks, which were arranged at short notice.