Gavin Williamson has hit back at the “haphazard way” the Huawei leak inquiry was carried out, complaining to friends he was only fired because of information he had personally volunteered to investigators working for Downing Street.

The former defence secretary showed the inquiry call records on his phone, which revealed that he had spoken to a Daily Telegraph journalist for 11 minutes from 5.31pm, shortly after a national security council meeting on Tuesday last week.

But while the ex-minister acknowledges being asked about what happened at the NSC, Williamson insists he told him nothing – and adds that his own admission of the phone call shows that he cannot be guilty.

Nevertheless, the fact of the call was the “compelling evidence” cited by the prime minister in his dismissal, according to the former minister, who believes he was the victim of a witch-hunt led by the cabinet secretary, Sir Mark Sedwill.

However, other government sources disputed Williamson’s account, saying the phone call with the Telegraph was not the sole basis of the “compelling” case against him. They added that the prime minister, not Sedwill, considered a series of pieces of evidence and made the judgment about how to proceed herself.

Williamson was interviewed for two hours last Friday, and thinks that he was unfairly targeted in a process where “scraps of paper” were used as evidence. By contrast the other eight ministers at the NSC meeting were only interviewed for “20 to 30 minutes” a time.

Another source said there was evidence that the Telegraph journalist made a series of calls to other government figures after speaking to Williamson, creating a theory that the reporter was seeking to confirm what the then minister had told him in the 11-minute conversation.

The NSC meeting had held a contentious discussion about whether to allow China’s Huawei to supply “non-core” 5G mobile phone equipment in the UK.

The evening after the meeting broke up, the Telegraph carried a report saying that five ministers, including Williamson, had raised objections about the proposal but the decision was in the end pushed through.

As he battled to clear his name, Williamson argued that the police should launch an investigation so he could be “absolutely exonerated”. But that appeared increasingly unlikely on Thursday, despite demands from Labour for a criminal inquiry into the NSC leak.

Responding to an urgent question from the opposition, David Lidington, the Cabinet Office minister, told MPs: “The prime minister has said she now considers that this matter has been closed and the cabinet secretary does not consider it necessary to refer it to the police.”

Cressida Dick, the Metropolitan police commissioner, added that any formal investigation would have to involve a referral from the Cabinet Office.

“It is sitting in the government. We, as the police, when considering whether there is an appropriate criminal investigation or not will have to be party to that material. At the present time we are not in possession,” she said.

Williamson referred to his “tough week” on Thursday, when he posted a picture of himself on Instagram with his dogs.

Williamson was to head back to Wolverhampton to join his family, where he was to consider his next steps. He has expressed the hope that the Daily Telegraph journalist he spoke to – Steven Swinford – would “do the right thing” and reveal his side of their conversation.

A Downing Street source said May had to move against a cabinet leaks culture that was threatening to overwhelm her premiership, and expressed the hope it would kill leaks from the NSC “stone dead”.

“The general mood among the cabinet is that this is a sorry affair, a sad business, and Gavin has lost a job he loves. But the evidence was there of a gradual build-up of leaks which started as tittle-tattle after cabinet meetings and gradually built up to serious leaks from NSC meetings.”

However, concerns about Huawei are unlikely to go away. Next week Mike Pompeo, the US secretary of state, is due to visit London, where he is expected to repeat the Trump administration’s warnings about using the firm’s technology in any part of the 5G network because of concerns it could be exploited if the Chinese government demands covert access.

Labour’s deputy leader, Tom Watson, said the government was trying to shut down legitimate questions about possible criminal activity by a minister before a police inquiry had been launched.

“The essential point here is the prime minister has sacked the secretary of state for defence because she believes there is compelling evidence that he has committed a crime. But despite that, she does not believe he should face a criminal investigation – where is the justice in that?”

Among Tory MPs who complained about the way May treated Williamson was Sir Edward Leigh, who said: “In this country we believe in natural justice ... As a matter of natural justice, how is the former defence secretary now going to be given an opportunity to prove his innocence?”

It also emerged that Williamson will be given a lump sum of around £17,000 after his dramatic removal from the government. As a former defence secretary, Williamson is entitled to the normal severance settlement enjoyed by any cabinet minister, No 10 said, which is three months of his annual salary of £67,505.

Dave Penman, the head of the FDA union representing senior civil servants, defended the decision to launch a full leak inquiry, insisting that similar inquiries have forced senior civil servants to hand over mobile phones on several occasions when they were not dealing with top secret information.

“Gavin Williamson, unlike a civil servant who had been found to have briefed a journalist after a national security council meeting, still has a job, plus a £17,000 payoff and a seat on the privy council.”