David Cameron says James Murdoch "clearly" needs to answer questions from MPs after his evidence on phone hacking was challenged.
Labour's Tom Watson wants a police probe after the evidence was disputed by two ex-News of the World executives.
The News International chairman had said he was not "aware" of an email suggesting hacking went beyond a single "rogue" reporter at the firm's paper.
He has now written to an MPs' committee to say he stands by his testimony.
Mr Murdoch appeared before the Commons culture, media and sport select committee on Tuesday alongside his father Rupert Murdoch, chairman of NI's parent company News Corporation.
Former NoW editor Colin Myler and NI legal manager Tom Crone maintain they "did inform him" about the email.
Following a statement issued by Mr Myler and Mr Crone on Thursday, committee chairman John Whittingdale said Mr Murdoch had agreed to write to them on various points he had been unable to address at the hearing.
He said: "I'm sure if the statement suggests there's conflict between what Colin Myler is saying and what he said, we will ask him to answer that as well."
In a letter sent to Mr Whittingdale, James Murdoch said he was in the process of preparing his written response.
He added: "Allegations have been made as to the veracity of my testimony to your committee on Tuesday. As you know, I was questioned thoroughly and I answered truthfully. I stand by my testimony."
Speaking during a visit in Warwickshire, the prime minister said: "Clearly James Murdoch has got questions to answer in Parliament and I'm sure he will do that.
"And clearly News International has got some big issues to deal with and a mess to clear up, that has to be done by the management of that company."
Labour leader Ed Miliband said: "People will want to look at the comments that were made and want to resolve the different versions of events that we've seen."
In April 2008, James Murdoch authorised the payment of an out-of-court settlement of more than £600,000 to Gordon Taylor, chief executive of the Professional Footballers' Association, over the hacking of his phone.
Mr Watson said he was going to formally bring the matter of James Murdoch's disputed evidence to the attention of Deputy Assistant Commissioner Sue Akers, who is leading Operation Weeting, the investigation into phone hacking.
The Metropolitan Police later confirmed Mr Watson's request for an investigation into the disputed evidence "was being considered".
The West Bromwich MP told the BBC: "This is the most significant moment of two years of investigation into phone hacking.
"If [Colin Myler and Tom Crone's] statement is accurate it shows James Murdoch had knowledge that others were involved in hacking as early as 2008, it shows he failed to act to discipline staff or initiate an internal investigation, which undermines Rupert Murdoch's evidence to our committee that the company had a zero tolerance to wrongdoing."
The MP added: "More importantly, it shows he not only failed to report a crime to the police but because there was a confidentiality clause involved in the settlement it means that he bought the silence of Gordon Taylor and that could mean he is facing investigation for perverting the course of justice."
He said: "There is only going to be one person who is accurate. Either James Murdoch, who to be fair to him is standing by his version of events, or Colin Myler and Tom Crone."
In other developments:
- undefined claims of phone hacking and breaches of data protection in Scotland. Its inquiry will centre on allegations that witnesses gave perjured evidence in the perjury trial of ex-MSP Tommy Sheridan
- John Yates, who resigned from his role as assistant commissioner of the Metropolitan Police on Monday, has issued High Court proceedings for libel against the London Evening Standard over aspects of its reporting of his conduct in the phone-hacking investigation
- The Law Society is to write to the judge leading the inquiry into phone hacking to investigate after revealing that police have warned solicitors that their phone messages might have been hacked
- The Solicitors Regulation Authority has launched its own investigation into the role of solicitors in the events surrounding the crisis
- Labour MP Chris Bryant, who is suing over allegations his phone was hacked, has written to non-executive directors of News Corporation asking for James and Rupert Murdoch to be suspended by the company's board
- Deputy Prime Minister Nick Clegg has confirmed he did raise questions about Mr Cameron's decision to bring ex-NoW editor Andy Coulson into No 10 as his director of communications
- The BBC has learned undefined following claims his mobile phone was hacked during a visit to the US. News International denies the claims.
In 2007, the News of the World's royal editor Clive Goodman and private investigator Glenn Mulcaire were jailed for hacking.
But the email referred to at the select committee was marked "for Neville" and Mr Watson's line of questioning was believed to be an attempt to see whether it implied that the News of the World's chief reporter Neville Thurlbeck also knew about such practices.
At the hearing, Mr Watson asked Mr Murdoch: "When you signed off the Taylor payment, did you see or were you made aware of the full Neville email, the transcript of the hacked voicemail messages?"
Mr Murdoch replied: "No, I was not aware of that at the time."
He went on: "There was every reason to settle the case, given the likelihood of losing the case and given the damages - we had received counsel - that would be levied."
In their statement issued on Thursday Mr Myler and Crone said: "Just by way of clarification relating to Tuesday's CMS select committee hearing, we would like to point out that James Murdoch's recollection of what he was told when agreeing to settle the Gordon Taylor litigation was mistaken.
"In fact, we did inform him of the 'for Neville' email which had been produced to us by Gordon Taylor's lawyers."