David Cameron is under pressure to release private emails exchanged with Rebekah Brooks that he has withheld from the Leveson Inquiry, after extraordinary scenes in the House of Commons in which he refused to explain the messages to MPs.

A visibly agitated Mr Cameron was asked during Prime Minister's Questions to tell the Commons why he had not disclosed details of secret messages between him and Ms Brooks, the former News International chief executive, or publicly revealed their existence.

In defiance of parliamentary convention, he flatly refused to tell MPs any more about the messages, which he has not supplied to the Leveson Inquiry into press ethics.

The Independent disclosed this week that Downing Street was sitting on a cache of emails and text messages between the Prime Minister and Ms Brooks, as well as communications with Andy Coulson, the former News of the World editor. Mr Cameron held them back from the judicial inquiry after taking legal advice that they were not relevant to its remit.

In acrimonious Commons clashes, the Prime Minister lost his cool as he was challenged by a shadow minister over the disclosures - and then said he would not answer any questions from the MP.

Last night the deputy Labour leader, Harriet Harman, sought to increase the pressure on Mr Cameron by urging him to release the information.

Sources within the inquiry have confirmed to The Independent that it has received none of the Brooks-Cameron emails. Downing Street has stonewalled on the subject, not denying the existence of the messages, and repeating that the Prime Minister has co-operated fully with the inquiry.

The relationship between Mr Cameron and Ms Brooks was closely scrutinised by the inquiry, as were the Prime Minister's dealings with Mr Coulson, who became his media chief in 2007. Mr Coulson has been charged with conspiring to hack phones and perjury. Ms Brooks is awaiting trial on charges of conspiring to hack phones and conspiring to pervert the course of justice. She denies the charges and Mr Coulson denies any knowledge of phone hacking.

During a fractious session in the Commons, Chris Bryant, the shadow Home Affairs minister, asked Mr Cameron: "Why won't you publish all the texts, emails and other forms of correspondence between yourself and your office and Rebekah Brooks, Andy Coulson and News International so we can judge whether they are relevant? Is it because they are too salacious and embarrassing for you... or is it because there is one rule for the Prime Minister and another for the rest of us?" Before he delivered his question, Mr Bryant told Mr Cameron to "stop smiling" and added: "When the truth comes out, the Prime Minister won't be smiling."

Mr Cameron retorted: "Before answering this question, I would like everyone to recall you stood up in this House and read out a whole lot of Leveson information that was under embargo and you were not meant to read out, much of which turned out, about me, to be untrue, and you have never apologised. Until you apologise, I'm not going to answer your questions."

He was referring to an occasion in April when Mr Bryant, who received Leveson evidence because as a hacking victim he was a "core participant", quoted from documents submitted to Lord Justice Leveson which had not at the time been published.

Ms Harman wrote to the Prime Minister last night urging him to release details of all electronic contacts with senior News International figures. She said: "In order for the public to have total confidence it would be preferable for you to disclose all of the emails and let the Leveson Inquiry decide which are relevant."