Mr Cummings was a campaign director for Vote Leave. He is now and advisor to Boris Johnson. According to one aide: “He was firm – if you leak you are gone.”
Another source claimed that he told any aides planning to take him to an employment tribunal: “You will be dead to me.”
Mr Cummings did not respond to any Telegraph requests for comment.
A source from Number 10 did say: “It sounds like someone is breaking the one strike policy. There has been a very clear message to special advisers that our focus is about delivering not about swanning around.”
Mr Johnson had said during his leadership campaign: “If you are going to have a discussion with serious senior colleagues, then you have got to allow people to speak their minds and then don’t leak it.”
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Mr Cummings is also believed to have told attendees at the meeting to prepare for an emergency budget in the week commencing October 7.
If this budget were to occur, it would be the first under new Chancellor Sajid Javid.
The last emergency budget came on June 22, 2010 under then new Chancellor George Osborne.
Leaks have been an in issue for the Government in recent months.
Sir Kim Darroch ended up resigning as US ambassador after leaked memos critical of US President Donald Trump were published in the national press.
Gavin Williamson lost his place as Defence Secretary in May after leaks over a National Security Council meeting.
Mr Williamson, now Education Secretary, has denied he was the leaker and insisted he would be exonerated by a full inquiry.
Though he was replaced by Penny Mordaunt, Mr Johnson’s cabinet changes saw Miss Mordaunt replaced by Ben Wallace.