Video report by ITV News Political Correspondent Carl Dinnen
The prime minister called for an independent review into how government contracts are secured, following complaints about Mr Cameron's conduct when lobbying ministers to win his employer financial support.
Downing Street said the PM wants the probe into Greensill Capital to be completed "thoroughly" and "promptly" and called for the review due to "significant interest" in the matter.
Mr Cameron, who was prime minister from 2010 to 2016, is under fire after newspaper investigations revealed he had personally texted chancellor Rishi Sunak to request access to an emergency coronavirus loan scheme for his employer.
Mr Cameron has denied breaking any lobbying rules, but admitted in a lengthy statement that he should have communicated with the government "through only the most formal of channels".
Despite being unable to win financial support for his employer, which later went bust, the former prime minister's correspondence with Mr Sunak and at least three other ministers while lobbying on behalf of Greensill Capital has raised questions.
There are also concerns about how the firm was able to win a number of supply chain finance contracts from the government while Mr Cameron was PM.
ITV News Political Correspondent Carl Dinnen with the latest from Downing Street
The Prime Minister's official spokesman told a Westminster briefing: "The Cabinet Office is commissioning an independent review on behalf of the Prime Minister, to establish the development and use of supply chain finance and associated activities in Government, and the role Greensill played in those.
"As you know, there is significant interest in this matter, so the Prime Minister has called for the review to ensure Government is completely transparent about such activities and that the public can see for themselves if good value was secured for taxpayers money.
"This independent review will also look at how contracts were secured and how business representatives engaged with Government."
Newspaper reports have also revealed Mr Cameron had arranged a "private drink" between Health Secretary Matt Hancock and Mr Greensill to discuss a payment scheme later rolled out in the NHS.
After reflecting "at length" about the allegations against him, Mr Cameron said he accepted there are “important lessons to be learnt”.
He said that "ultimately" the outcome of his efforts to get Greensill's proposals included in the Government's Covid Corporate Financing Facility (CCFF) was that, "they were not taken up".
"So, I complied with the rules and my interventions did not lead to a change in the Government's approach to the CCFF," Mr Cameron added.
"However, I have reflected on this at length. There are important lessons to be learnt.
"As a former prime minister, I accept that communications with government need to be done through only the most formal of channels, so there can be no room for misinterpretation."
But his 1,300 word statement addressing the matter was not enough to satisfy Prime Minister Johnson.
The PM's spokesman said: "As you would expect, the Prime Minister wants this to be done thoroughly and he wants it to be done promptly. So you can expect a prompt return on this."
He added that the independent review would be led by legal expert Nigel Boardman and that the Cabinet Office will set out further formal details in due course.
Downing Street said Mr Boardman would have "access to the documents that he needs" for his probe.
But Labour said the review risks kicking the issue into the "long grass".
Shadow chancellor of the Duchy of Lancaster Rachel Reeves said: "This has all the hallmarks of another cover-up by the Conservatives.
"Just as with the inquiry into Priti Patel's alleged bullying, this is another Conservative Government attempt to push bad behaviour into the long grass and hope the British public forgets.
Listen to the ITV News Politics Podcast:
"We need answers on Greensill now - that means key players in this cronyism scandal like David Cameron, Rishi Sunak and Matt Hancock appearing openly in front of Parliament as soon as possible to answer questions."
Mr Cameron said that "many of the allegations" made in recent weeks "are not correct" and he challenged the "false impression" that Mr Greensill was a key member of his team while in No 10.
Prime Minister Johnson's spokesman, asked whether Mr Johnson believed lobbying rules needed to be changed, said: "As you have seen from what we have announced today, the Prime Minister understands the significant public interest in this and wants to look at the issues raised and get more details.
"But I think you can judge from his actions."