The Prime Minister is “contrite” over allegations of Covid rule-breaking and will seek to “address the underlying culture in Downing Street” that led to partygate, a Cabinet minister said.

Conservative Party chairman Oliver Dowden admitted there were “failings” in No 10, following a series of leaks about alleged lockdown parties, but he denied it was a resigning matter for Boris Johnson.

Mr Dowden said the Government plans to “address the kind of culture that has allowed” the reported flouting of coronavirus laws to happen, in a hint of a shake-up at the top of Mr Johnson’s administration.

It comes after The Sunday Times reported that the Prime Minister is devising a policy announcement blitz and a cull of his inner circle as he looks to survive the publication of senior civil servant Sue Gray’s report into so-called partygate.

Martin Reynolds, his principal private secretary who sent an email inviting staff to “bring your own booze” in the No 10 garden during the first coronavirus lockdown, and Mr Reynolds’ deputy Stuart Glassborow are likely to be forced out of Downing Street, according to the newspaper.

No 10 chief of staff Dan Rosenfield’s position could also be at risk, it suggested.

Speaking to the BBC’s Sunday Morning programme, Mr Dowden said: “I can assure you the Prime Minister is both very contrite and deeply apologetic for what happened.

“But, more importantly, he is determined to make sure that this can’t be allowed to happen and that we address the underlying culture in Downing Street.

“There were failings: we should have done better, much, much better.”

It comes as a sixth Conservative MP called for Mr Johnson to quit, arguing that a change of senior officials would not reverse the “terminal damage” done to the Prime Minister’s reputation by the allegations.

Former children’s minister Tim Loughton, in a post published on Facebook on Saturday, said: “It is not down to a simple Government policy change or a sacking of ministers or officials to put things right.

“In this case all roads lead back to Downing Street and the person whose name is on the front door.”

Former Tory leader Sir Iain Duncan Smith was also heavily critical of the No 10 regime, labelling the possible lockdown breaches “unforgivable” and blaming it on a “lazy and slack” culture in Downing Street.

Sir Iain told Sky News’ Trevor Phillips On Sunday programme that Cabinet Office official Ms Gray’s report would settle the question of whether the Prime Minister “knew or understood what was going on”.

However, veteran Tory Peter Bone told LBC he had found constituents in his Wellingborough seat were “clearly in support of the Prime Minister” while former trade secretary Dr Liam Fox – who was sacked by Mr Johnson – said it was the “wrong time” for a change of leader.

For a leadership contest to be triggered, 54 letters of no confidence in Mr Johnson have to be submitted by Tory MPs to Sir Graham Brady, the chairman of the 1922 Committee, asking for a vote on the Conservative leader’s future.

Sir Graham does not publicly state how many letters he has received, but reports suggest about 20 might have been handed in, with East Worthing and Shoreham MP Mr Loughton indicating he could be persuaded to put his own thoughts into writing should the Prime Minister not quit in the “next few days”.

The Sunday Times reported Mr Johnson will look to get on the front foot and save his position, having last week admitted attending the No 10 garden party on May 20 2020, by making a series of “populist” announcements in the coming weeks.

He will focus on reducing the NHS backlog and tackling the small boat crossings in the Channel, while freezing the BBC licence fee for two years, it was reported.

The Conservative Party leader could also put in place a “booze ban” in No 10 following the series of embarrassing claims of Covid rule-breaking.

Separately, questions were raised about whether Mr Johnson had been totally upfront in his apology to the Commons on Wednesday, in which he said he attended the May 20 garden party for 25 minutes but understood it to be a “work event”.

A report in the Sunday Times has claimed the Prime Minister was warned by “at least two people” that the event being organised in Mr Reynolds’ email amounted to “a party” and should be immediately cancelled.

Mr Johnson is said, according to the newspaper, to have dismissed the concerns as an overreaction,  praising his private secretary as “my loyal Labrador”.

Deputy Labour leader Angela Rayner said the report “blows the Prime Minister’s final defence out of the water”, but Downing Street denied its veracity.

A No 10 spokeswoman said: “It is untrue that the Prime Minister was warned about the event in advance.

“As he said earlier this week he believed implicitly that this was a work event.

“He has apologised to the House and is committed to making a further statement once the investigation concludes.”

Meanwhile, Labour leader Sir Keir Starmer repeated his assertion that Mr Johnson should resign, arguing he had “degraded” the office of prime minister and “lost all authority”.

Sir Keir said his rival “broke the law” and recommended that Ms Gray’s findings be “passed to the police to look at” once finalised.

The Metropolitan Police have said they will hold back on investigating until the review is over.

Shadow health secretary Wes Streeting told Sky a no confidence vote in Parliament would only “galvanise” the Tories.

But the Liberal Democrats have found cross-party support for a motion of no confidence in the Prime Minister, including from two Labour MPs, Paula Barker and Mick Whitley.

Lib Dem leader Sir Ed Davey encouraged Tory MPs to back the motion, adding: “The country deserves a chance to move on from this deceitful Prime Minister.”