The shadow defence secretary, Jim Murphy, has written to the prime minister demanding a wider inquiry into the extent of the links between the defence secretary, Liam Fox, and his self-styled adviser, Adam Werritty.

He says the current inquiry, established by Fox himself and due to report to the prime minister on its initial findings on Monday, is inadequate and too narrow.

Murphy urges that any breach of the ministerial code should be referred to the independent adviser on ministers' interests, Sir Philip Mawer.

Murphy also claimed that a number of Fox's assurances about his connections with Werritty, including some given to him personally, now appeared to be "at variance with the evidence".

In his letter Murphy claims: "To arrive at a meaningful judgment on whether the ministerial code has been breached it is necessary to assess all the issues that have been raised. As you will know, the 2010 ministerial code states explicitly that it is not the role of officials to enforce the code and it is therefore inappropriate for the permanent secretary to undertake this role.

"The code states that if there is an allegation about a breach then you as prime minister should refer this to the independent adviser on ministers' interests. This course of action is now clearly necessary and I urge you take it immediately."

Murphy adds that Fox "said on 15 September in parliamentary answers that Mr Werritty had 'not travelled with me on any official overseas visits'. The MoD said on 7 October that Mr Werritty 'did not attend any official meetings'. New evidence suggests otherwise. Dr Fox told me in a conversation on 13 July that the civil service were present at the Dubai meeting on 17 June. Again, available evidence appears to contradict this."

The letter goes on: "The secretary of state also claimed to the BBC yesterday that the meeting in Dubai happened by accident when 'they happened to be sitting at a nearby table at a restaurant'. Again, this assertion does not seem to be supported by available evidence.

"There are numerous emails from Mr Werritty making arrangements for the meeting to take place. Importantly, this meeting reportedly discussed how a technology could serve both troops in Afghanistan and be of use in Libya, as well as an ongoing legal case involving the MoD. These are both issues which should not be discussed without MoD officials present."

Fox will be forced to clarify his relationship with Werritty when he answers a long scheduled session of defence questions in the Commons on Monday.

Although Murphy said he would be urging the Speaker, John Bercow, to require Fox to make a full statement in the Commons, Fox is already due to face Murphy at defence questions.

It is likely Downing Street will want a clear line from the hurried inquiry set up by Cameron on Saturday before defence questions start at 2.30pm.

Cameron asked the cabinet secretary, Gus O'Donnell, to have an initial report on his desk by Monday, so short-circuiting the more leisurely, fortnight-long internal inquiry initially set up by Fox himself.

Cameron will want to have a clear sense of Fox's culpability by the time he answers any likely questions from the media.

Labour has refrained from calling for Fox's resignation, but the deputy leader, Harriet Harman, piled on the pressure, saying there had to be a full statement on Monday. "There's now a further question about Dr Fox's integrity and the question of whether he actually answered truthfully and fully all the questions that have been put."

Werritty is not employed by the MoD and does not have national security clearance.

Emails and video footage have emerged appearing to contradict Fox's explanation of the meetings. A video has emerged showing that Werritty, who has no official role, attended a meeting between the Sri Lankan president and the defence secretary.

Emails suggest a meeting with Dubai businessmen was not a chance encounter, as he had claimed.

The former prime minister Sir John Major, a veteran of previous cabinet scandals, told the BBC's Andrew Marr Show: "It's extremely difficult to handle these. Either natural justice requires you wait a long time and you're then said to be incapable of making decisions if you operate under natural justice, or you move too speedily and you're said to be ruthless.

"So I don't know what the situation is. The prime minister has asked for the facts to have a first look on Monday and I think that is right."

The energy secretary, Chris Huhne, urged the media not to rush to judgment on Fox.

Overall Fox is seen to have done a good job at defence and Downing Street will be reluctant to lose him, partly since he is a standard bearer of the party right, and someone whom party modernisers would not like to see roaming on the backbenches with a licence to criticise.