The BBC said the documents, almost 50 pages in all, were found "in a soggy heap behind a bus stop in Kent early on Tuesday morning" by a member of the public, who wanted to remain anonymous.
The Daily Telegraph suggested the report's findings could be comparable to phone-hacking revelations at the News of the World tabloid, which forced its closure.
The probe is likely to be independent and carried out on behalf of the Cabinet Office, the BBC said. Reuters has not been able to independently verify the reports.
In his first comments on the row on Sunday, Cameron said that in his representations to government he was breaking no codes of conduct and no government rules.
“John’s work has exposed truth the Chinese authorities did not want the world to know,” BBC News said. “The BBC is proud of John’s award-winning reporting during his time in Beijing and he remains our China correspondent.”
During the Singapore summit, Trump gave Kim a glimpse inside his presidential state car -- a $1.5 million Cadillac also known as "The Beast" -- in a show of their newly friendly rapport.