Mr Cameron has come under fire for lobbying on behalf of the recently collapsed finance firm and this in turn has sparked a wider row over private companies' attempts to influence ministers and officials.
The government has asked a senior lawyer to conduct a review into the issue and to report back by the end of June.
However, some MPs feared this probe might not stringent enough and on Wednesday the Commons Treasury Select Committee announced plans for its own probe, with other committees reportedly planning to do the same.
A spokesperson for Mr Cameron has said he would respond “positively” to any request to give evidence to any of the inquiries that are taking place once the terms of reference have been established.
On Thursday, the government’s links with Greensill will come under further scrutiny when the head of the watchdog which exposed the way the former head of government procurement was able to take a position as a part time adviser to the firm while still in Whitehall, is to appear before MPs.
Lord Pickles, the chair of the Advisory Committee on Business Appointments (Acoba) will give evidence on Thursday to the Commons Public Administration and Constitutional Affairs Committee (PACAC) amid “acute concern” at the latest developments.
The former cabinet minister had sought an explanation as to why Bill Crothers had taken a job with Greensill without clearing it with Acoba which is supposed to vet all such appointments by former ministers and senior officials.
To his apparent astonishment he was told by the Cabinet Office that it was unnecessary as Mr Crothers was already working for the firm and so the appointment did not have to be declared.
In response, the Cabinet Secretary Simon Case has called on all Whitehall departments to notify him of any senior officials with paid jobs outside of government by the end of the week.
In a letter to permanent secretaries, Mr Case said the disclosures were a cause of “acute concern” and stressed the need for the Civil Service to “engage fully” with the review commissioned by Prime Minister Boris Johnson by senior lawyer Nigel Boardman.
The hearing with Lord Pickles comes as PACAC chairman William Wragg indicated the committee could launch a wider investigation into the Greensill affair.
Meanwhile a second group of MPs on the Commons Treasury Committee has announced it is to mount an inquiry into the firm’s collapse which threatens thousands of UK jobs at Liberty Steel which relied on the firm’s financial backing.
The move came after the government used its Commons majority to defeat an attempt by Labour to force the creation of a new committee of MPs specifically to examine the issues of lobbying and the Greensill affair.
It followed disclosures that Mr Cameron personally lobbied Chancellor Rishi Sunak on its behalf and was able to arrange for its founder, Lex Greensill, to have a “private drink” with Health Secretary Matt Hancock.
Mr Cameron has insisted he did not break any rules but acknowledged there were “lessons to be learned” and that, as a former prime minister, any contacts with government should be through the “most formal channels”.
In the Commons on Wednesday, Mr Johnson acknowledged that the affair had raised concerns, suggesting the boundaries between government and business had not been “properly understood”.
Labour leader Sir Keir Starmer went on the offensive saying it marked the return of “Tory sleaze” with “dodgy contracts, privileged access, jobs for their mates” and an “open door” between the Government and business.
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