The disclosure came after Downing Street said it would not be mounting its own inquiry into the claims, despite calls to do so by both Conservative and opposition MPs.
A No 10 spokesman said it would only open an inquiry if it was presented with evidence to back up Mr Wragg’s assertions.
However the MP, who chairs the Commons Public Administration and Constitutional Affairs Committee, said he believed an investigation should be for the “experts” in the police.
He told The Daily Telegraph that he would outline “several” examples of bullying and intimidation, in some cases involving public money.
“I stand by what I have said. No amount of gas-lighting will change that,” he told the newspaper.
“The offer of Number 10 to investigate is kind but I shall leave it to the experts. I am meeting the police early next week.”
Former Tory MP Jerry Hayes threw his support behind Mr Wragg and said the claims should be investigated.
Mr Hayes told the BBC: “We’ve got William Wragg, who is actually a serious figure, he’s not a snowflake, he’s not someone who can easily be bullied.
“Something has happened and he’s gone all the way to the Metropolitan Police. I just hope that we have an investigation. If he’s gone to the Metropolitan Police there must be something there, otherwise his career falls apart, doesn’t it?”
A Metropolitan Police spokesman said: “As with any such allegations, should a criminal offence be reported to the Met, it would be considered.”
Mr Wragg’s latest intervention comes as No 10 is braced for the expected delivery next week of the report of Sue Gray, the senior civil servant investigating lockdown parties in Downing St elsewhere in Whitehall.
It is likely to lead to renewed calls from opposition parties for a police investigation if there is any evidence Covid rules were broken – including at a drinks do in May 2020 attended by Mr Johnson.
Mr Wragg, one of seven Tory MPs to have called publicly for the Prime Minister to resign, stunned Westminster with his allegations this week of a campaign of intimidation by No 10 amounting to criminal conduct.
He said Conservative MPs trying to trigger a no confidence vote in Mr Johnson had been told public funding for projects in their constituencies would be cut off and threatened with stories in the press to embarrass them.
Christian Wakeford, the Bury South MP who defected to Labour, later described how the Tory whips had warned him over funding for a new school in his constituency if he rebelled in a vote over free school meals.
Ministers have sought to dismiss the allegations, insisting the whips had no role in the allocation of public funding.
Business Secretary Kwasi Kwarteng however acknowledged on Friday that Mr Wakeford’s allegation was “very serious” and he was “sure” that it would be investigated.
But a spokesman for the Prime Minister said: “We’re not aware of any evidence to support what are clearly serious allegations.
“If there was any evidence to support it, it would of course be looked at.”
The latest disclosures will only fuel the febrile mood at Westminster, with Mr Johnson’s political survival hanging in the balance.
Mr Wakeford’s defection appeared to have put the plotting on hold as Tory MPs publicly rallied behind the leadership, while the rebels largely went to ground.
However the publication of Ms Gray’s report represents another moment of danger, potentially triggering a fresh wave of letters to the chairman of the Tory backbench 1922 Committee Sir Graham Brady.
Under party rules there will be a confidence vote in Mr Johnson if 54 of the party’s MPs write to Sir Graham calling for one.
Mr Johnson is expected to spend the weekend at Chequers, his official country residence, ringing round potential rebels urging them not to plunge the dagger.
The Times reported the Prime Minister had reassembled the ministerial team which helped him mount his successful leadership bid in 2019 as he seeks to shore up support.
Transport Secretary Grant Shapps is reportedly playing key role in the operation along with three former whips and other loyalists.