The SNP’s shadow Cabinet Office spokesperson Stewart Hosie has written to Mr Johnson demanding ministers disclose their full exchanges with the former prime minister after it emerged Mr Cameron lobbied the Chancellor and health secretary for access to the Greensill Capital to taxpayer funded Covid loan schemes.
Mr Cameron exchanged casual text messages with Rishi Sunak and took Matt Hancock to private drinks to discuss financial support for the firm.
In a letter to the Prime Minister, Mr Hosie claimed it "raises serious questions about the access granted to government departments to firms with close links to the Conservative party."
He said: "While around 2.4 million people and businesses were left without UK Government Covid financial support after a year of the pandemic, it turns out David Cameron had senior Cabinet ministers on speed dial and was casually exchanging text messages and arranging private drinks with them to access Covid cash and contracts.
"The serious revelations of the sustained lobbying of government ministers has yet again exposed the rampant cronyism at the heart of Boris Johnson's Tory government.
"Given the seriousness of the issue and the litany of examples of Tory cronyism, it is vital that Rishi Sunak and Matt Hancock come before Parliament to explain their conduct and disclose their full exchanges with David Cameron.
"The Tories have form in attempting to kick scandals into the long grass, so while the belated announcement of an inquiry is welcome, it's clear we need answers from Cabinet ministers now.
“They cannot use the inquiry as a cheap excuse to duck accountability.”
The former prime minister has accepted he should have communicated with the UK Government “through only the most formal of channels” rather than text messages.
No.10 has confirmed a review will be launched into Greensill.
Lawyer Nigel Boardman is due to lead the probe, which will examine how government contracts were secured by the company as well as the actions of Mr Cameron.
Downing Street said Mr Johnson had called for the independent review to be launched into Greensill due to "significant interest" in the matter.
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."
A spokesman for Mr Cameron said: "We welcome this inquiry and will be glad to take part."
The failure of Greensill, a financial services company, has left many fearing that thousands of jobs at Liberty Steel’s assets in the UK could be at risk, including those who work at the Lochaber aluminium smelting plant.
A senior UK Government source said the SNP had their own "questions to answer" about Greensill.