Cameron, who quit as prime minister in 2016 in the wake of the Brexit referendum, on Sunday gave his first response since his lobbying efforts for Greensill were made public, saying he had not broken any codes of conduct or lobbying rules.
Gordon Brown, U.K. prime minister from 2007 to 2010, on Monday called for a five-year ban on lobbying by former ministers.
The firm later collapsed, putting thousands of United Kingdom steelmaking jobs at risk. This was to discuss a payment scheme later rolled out in the NHS.
He also arranged a private meeting with Health Secretary Matthew Hancock, after which a Greensill payment programme was used in the National Health Service, The Sunday Times reported.
The idea was to pay doctors and nurses either daily or weekly.
A Department of Health and Social Care (DHSC) subsidiary subsequently announced in October 2020 that mobile app Earnd, then a division of Greensill, would be available free-of-charge to NHS employees to access their pay.
Brown said the government must act quickly in response to the Greensill affair because it has the potential to bring public service into "disrepute", just like the parliamentary expenses scandal of 2008.
Mr Cameron and government ministers have come under fire in recent weeks for their connections to Greensill, which specialised in financing supply-chain invoices but collapsed last month.
Sir John Major's intervention comes after the former Labour Prime Minister Gordon Brown called for tougher rules Credit Jeff Overs BBC via Reuters
"Our approach was and is that local NHS employers are best placed to decide how different pay flexibilities fit with their overall pay and reward offer for their staff".
Separately, it was understood that Cameron's message to the prime minister's adviser was forwarded on to the Treasury.
But it could not be immediately confirmed whether the lobbying did lead to the Treasury reconsidering its move to reject the loan scheme application.
Throughout the pandemic, an huge number of businesses contacted Downing Street with representations; these were passed on to relevant departments.
Mr Cameron was said to have described the decision to exclude his employer's firm, Greensill Capital, from the multibillion-pound scheme as "nuts" and pressed for the Chancellor to reconsider.
It had been reported that a proposal from Mr Greensill, while he worked in No10, for NHS-affiliated pharmacies to be paid using private finance was "handed directly to Mr Cameron, who signed it off. bypassing Francis Maude entirely".
Labour said Mr Sunak has been absent from parliament since 9 March.
Shadow chief secretary to the treasury Bridget Phillipson said: "Every day brings fresh revelations about the culture of cronyism at the heart of this Conservative Government".
The government inquiry sparked by the furore over David Cameron's lobbying on behalf of Greensill is expected to have three "buckets", according to sources. But Bernard Jenkin, a lawmaker who led an inquiry into links between government and business, said the only way to combat this long-running problem is to require serving ministers and civil servants to report inappropriate conduct by lobbyists.
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