The firm collapsed into insolvency last month.
It was also reported that the former British prime minister approached the Bank of England in an attempt to receive money for the company.
The former prime minister has been criticised for contacting ministers via text on behalf of the company, which collapsed in March. At the time, Greensill's firm wanted to introduce a scheme to pay doctors and nurses flexibly via its mobile app Earnd.
NHS SBS, a joint venture between the Department of Health and Social Care (DHSC) and a French IT firm, went on to announce in October a year ago that Earnd, a mobile app that was then a division of Greensill, would be available free-of-charge to NHS employees to access their pay. Hancock reportedly asked Greensill to work directly with the NHS rather than the DHSC.
The BBC, in turn, cited an unnamed Hancock ally as saying that the health secretary "acted in entirely the correct way", and that he "updated officials on the business that was discussed, as is appropriate".
He anxious that the scandal may sully the public's trust in politicians.
Cameron, who was employed as a part-time adviser to Greensill, said his work on behalf of the company didn't break any rules or codes of conduct on the activities of former ministers.
Allies of the former PM - cleared of breaking lobbying rules - claim he has been caught in a "pincer movement" between No10 aides loyal to Mr Gove and former mandarins Lord Macpherson and Lord Kerslake, now crossbench peers.
The chancellor does not have to appear in parliament himself and a Treasury minister is expected to stand in for him. But it couldn't be immediately confirmed whether the lobbying did lead to the Treasury reconsidering Greensill's loan scheme application.
A No 10 spokesman: "Throughout the pandemic, an huge number of businesses contacted Downing Street with representations; these were passed on to relevant departments".
"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", he told reporters.
United Kingdom finance minister Rishi Sunak is said to have received text messages from Cameron on behalf of the now-insolvent finance firm.
"Mr Boardman will have access to all necessary government information required to conduct the review and will engage with those involved at the time when decisions were made".
It was allowed access to tens of millions of pounds of taxpayer money to issue loans under a separate pandemic support scheme known as the Coronavirus Large Business Interruption Loan Scheme (CLBILS).
Labour MP Dame Margaret Hodge, a former chair of the Commons Public Accounts Committee, said: "We need an independent inquiry immediately".