The former Prime Minister sent ministers and officials 45 emails, texts and WhatsApp messages relating to the now-collapsed firm in less than four months.
Mr Cameron also sent messages to vaccines minister Nadhim Zahawi as he sought to gain access to UK Government-backed coronavirus loans and dismissed opposition to it as “nuts” and “bonkers”.
Greensill was placed into administration in March, threatening thousands of UK jobs at Liberty Steel.
Texting Mr Gove, Mr Cameron said: “But do you have a moment for a word? I am on this number and v free. All good wishes Dc.”
The Cabinet Office minister replied: “Thank you! Will call!”.
On the same day, Mr Cameron texted the Chancellor asking for a “very quick word” to discuss the Treasury’s refusal to grant access to the Covid Corporate Financing Facility (CCFF).
He said: “HMT are refusing to extend CCFF to include supply chain finance, which is nuts as it pumps billions of cheap credit into SMEs.
“Think there is a simple misunderstanding that I can explain.”
After arranging a discussion with Mr Sunak, the former Tory prime minister messaged Mr Gove again.
He said: “Am now speaking to Rishi first thing tomorrow. If I am still stuck, can I call you then?”
On April 22, he texted Mr Sunak again asking if he could “give it another nudge over the finish line”.
Others contacted include Boris Johnson’s senior adviser Sheridan Westlake, Treasury ministers Jesse Norman and John Glen, and deputy Bank of England governor Sir Jon Cunliffe.
He also called UK health secretary Matt Hancock. .
Mr Cameron has previously insisted he broke "no codes of conduct and no government rules".
He said: "As a former prime minister, I accept that communications with the government need to be done through only the most formal of channels, so there can be no room for misinterpretation."
The emergence of the messages came as Lex Greensill apologised and said he takes “complete responsibility” for the collapse of Greensill Capital.
Appearing before the Commons Treasury Committee, he said: “Please understand that I bear complete responsibility for the collapse of Greensill Capital.
“I am desperately saddened that more than 1,000 very hard working people have lost their jobs at Greensill. Likewise I take full responsibility for any hardship being felt by our clients and their suppliers, and indeed investors in our programmes.
“It’s deeply regrettable that we were let down by our leading insurer whose actions assured Greensill’s collapse, and indeed some of our biggest customers.
“To all of those affected by this, I am truly sorry.”
SNP Westminster depute leader Kirsten Oswald claimed the messages showed a sense of "entitlement".
She said: “It speaks volumes that a former Tory Prime Minister thinks it’s acceptable to drop a text to the current Tory Cabinet members for back-door access to vital Covid schemes.
“This is the standard the Tory government has set – and perpetuated throughout the years.
"Despite the efforts of the SNP to stamp out the culture of sleaze engulfing this Tory government – whether it’s the current Greensill scandal [and] Covid contracts for cronies – it is clear that Westminster is broken beyond repair.”