David Cameron will respond "positively" to any request to give evidence to inquiries into the Greensill affair, according to his spokesman.
MPs are to further probe government links to the collapsed financial firm on Thursday when the chairman of watchdog the Advisory Committee on Business Appointments (Acoba) gives evidence.
Acoba is supposed to vet such appointments but the Cabinet Office said it was unnecessary as Mr Crothers started working for the firm before he left the Civil Service – so did not have to declare it.
Lord Pickles will be questioned on the details later by the Public Administration and Constitutional Affairs Committee (PACAC), whose chairman has suggested a wider investigation could be launched.
David Cameron started working for Greensill in 2018 and personally lobbied Chancellor Rishi Sunak on its behalf to try to get the firm access to a government loan scheme.
He sent multiple text messages to the chancellor, as well as arranging a "private drink" for billionaire founder Lex Greensill with the health secretary.
The former PM, who led the country until 2016, has insisted he did not break any rules but admitted there were "lessons to be learned". He also conceded that his contact with government should have been via the "most formal channels".
The fallout from the Greensill affair has led to an urgent order for senior civil servants to declare any other jobs they hold outside government by the end of the week.
Cabinet Secretary Simon Case said there was "acute concern" about issues that have emerged in the last few weeks.
Mr Case sent a letter to permanent secretaries saying there is a role in government for people with outside expertise but that it was essential for the "integrity and impartiality" of the Civil Service to be maintained.
He said there must be "transparency and full proper management" of any outside interests.
MPs on the Commons Treasury Committee have also said they will start an inquiry into the firm’s collapse – in March this year – as it threatens thousands of jobs at Liberty Steel in the UK.
A Labour attempt to create a new committee specifically to focus on lobbying issues and the Greensill affair was defeated using the government’s Commons majority.
Boris Johnson admitted on Wednesday that the matter raised concerns and that the boundaries between business and government may not have been "properly understood".
Labour leader Sir Keir Starmer, speaking in the Commons, hit back at the PM and said it heralded the return of "Tory sleaze" with "dodgy contracts, privileged access, jobs for their mates".