LONDON -The role Cameron played in trying to lobby ministers on behalf of the failed finance company has raised questions about government access, particularly by former ministers who take up paid employment with private firms.
Johnson's spokesman said the review would not have legal powers but would have access "to all the necessary information required and engage with those involved at the time".
It would be able to make recommendations, he added.
Cameron has said he had not broken any lobbying rules, but that he accepted his communication with government should be completely formal.
The former prime minister has indicated he will take part in the review, but the main opposition Labour party has said the government's response does not go far enough.
"I think people have just got questions that they need to satisfy themselves, including me, about how this supply chain finance stuff is going to work," Johnson told reporters.
Asked what he made of the behaviour of his former boss Cameron, Johnson said: "That's a matter for Nigel," referring to Nigel Boardman, who will lead the investigation into Greensill.
He said Boardman would have "pretty much carte blanche to ask anybody whatever he needs to find out".
The review was prompted after the Financial Times and Sunday Times newspapers reported that Cameron contacted ministers on behalf of Greensill, including sending text messages to finance minister Rishi Sunak and arranging a drink between Australian banker Lex Greensill and Health Secretary Matt Hancock.
Lex Greensill was brought in as an adviser to the government while Cameron was British prime minister from 2010 to 2016. After leaving office, Cameron became an adviser to Greensill's now-insolvent company.
On Sunday, Cameron said in a statement to the Press Association he did not break any codes of conduct or government rules and noted that ultimately the outcome of the discussions on Greensill's proposals on a loan were not taken up.
But he did say that important lessons needed 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."
(Reporting by Paul Sandle, Sarah Young and Elizabeth Piper)