The powerful Treasury Committee has sent all the key individuals a series of questions about what happened, ahead of looming evidence sessions.
The inquiry is one of more than half-a-dozen launched since Mr Cameron’s direct lobbying of government ministers on behalf of Greensill, a failed finance firm, was revealed.
Mr Sunak, the chancellor, is in the spotlight after he told the former prime minister – in a private text – that he had “pushed” civil servants to help the company join a Covid-19 loan scheme.
Mel Stride, the committee’s Conservative chair, said: “There are questions to be answered in relation to Greensill Capital regarding the operation of the UK’s financial system and its regulation.
“Also, whether the Treasury responded appropriately to lobbying from Greensill during the pandemic.”
In his letter to Mr Sunak, Mr Stride has called for a full timeline of the contacts that Treasury officials and ministers had with Mr Cameron and other Greensill representatives.
Mr Cameron has been asked to produce the full texts that he sent the Chancellor, which the Treasury has so far refused to publish.
Boris Johnson responded to the scandal – which included procurement chief Bill Crothers working as a Greensill adviser while in his Whitehall job – by asking lawyer Nigel Boardman to investigate.
But that inquiry has been widely rubbished as not properly independent – Mr Boardman is a paid director of the business department – and criticised for a narrow remit that will fail to restore public confidence.
The Public Administration Committee yesterday will investigate whether existing lobbying rules and penalties are tough enough.
Announcing its terms of reference, it noted Greensill’s relationship with ministers and Whitehall had raised “significant concerns about the propriety of governance in this country” which “risks undermining public trust”.
“We will look into whether the rules need tightening up and clarifying and we will make any necessary recommendations without fear or favour,” said William Wragg, the committee’s Tory chair.
It comes after the cabinet secretary, Simon Case, ordered department chiefs to declare any instances of senior officials with second jobs outside the Civil Service.
Mr Johnson has sought to play down the extent of those links, insisting there are not “loads of people” working as civil servants while also employed by a business.
“People should not, in my view, form the impression that the upper echelons of the British Civil Service have got loads of people who are double-hatting, as it were, doing two jobs – it just isn’t true,” he insisted.
Earlier, the head of the committee on standards in public life urged the prime minister to give up the power to decide whether his ministers are investigated for sleaze.
The recommendation – which follows the failure to probe complaints against Robert Jenrick and Mr Sunak – came in a letter from Lord Evans.
He also said Mr Johnson should relinquish the right to decide whether a colleague has breached the ministerial code and hand that responsibility to an independent adviser.