Sue Gray is expected to deliver her report on No 10 parties to the PM without waiting for the police inquiry to conclude, the BBC has been told.
The senior civil servant is set to hand her report to Downing Street shortly - though no exact date has been given.
But police have asked for "minimal reference" to be made to events they are looking at, in order to "avoid any prejudice to our investigation".
Tory MP Sir Christopher Chope said the Met's actions were an "abuse of power".
Speaking to BBC Radio 4's Today programme, he accused the police of "interfering" with the content of the report and trying to prevent it being released in full.
Opposition parties have also insisted the report is released in full.
No 10 has promised to publish Ms Gray's findings on alleged lockdown-breaking parties in Downing Street and Whitehall, but is yet to receive the document.
Another Tory MP Adam Holloway, who said he backs Boris Johnson but is willing to "run with" Ms Gray's findings, told the same programme: "If I was Boris I would be really quite desperate for the truth to get out."
And Lord Morris of Aberavon, a former attorney general under ex-Labour PM Tony Blair, said he was dismayed with the police, telling the BBC: "Any prejudice which might result in fines would be a disproportionate concern."
But criminal barrister Matthew Scott, from Pump Court Chambers, said: "Given that they are investigating possible criminal offences I think they are absolutely right to do what they have done and ask.
"They have no power to enforce, but they can ask and they have done, that the relevant parts of the report not be made public before they have completed their inquiry."
Earlier this week, the Metropolitan Police said it was investigating the parties and gatherings.
The force later said it did not ask for the report to be delayed but was speaking to the Cabinet Office on the content of the report "to avoid any prejudice to our investigation".
This means the report Ms Gray releases before the police investigation is complete may need to contain some redactions, or be changed.
Mr Scott said the police's term "prejudice" did not refer to prejudicing a jury trial as there would not be one for any potential breach of lockdown rules.
Instead, the term referred to "spoiling their investigation".
Officers would not want witnesses or suspects to be influenced by what other people had told them already, he told BBC Breakfast.
"The police have no power to force Sue Gray to change her inquiry, or to redact her inquiry," he said, adding that the PM was also "perfectly at liberty" to publish the outcome - or indeed comply with a police request.
BBC political correspondent Iain Watson said he had been told Ms Gray was trying to redraft parts of her report to address any police concerns.
But, he said, she wanted to avoid blanking out whole swathes of text in case it looked like "a Whitehall whitewash".
Many Conservative MPs are saying they will wait for the publication before deciding whether to take action against the prime minister.
Material from the Cabinet Office was received by the force on Friday. And, in a statement, the Met said it expected to approach individuals "identified as having potentially breached" regulations.
Commander Catherine Roper, who leads the Met's Central Specialist Crime Command, said "the offences under investigation, where proven, would normally result in the issuing of a fixed penalty notice".
She added that the Met's actions would be "proportionate to the nature of these offences".
Former director of public prosecutions Lord Macdonald told the BBC that asking for Ms Gray's report to make only minimal reference to the gatherings being investigated by police seemed "disproportionate".
Tory MP Sir Geoffrey Clifton-Brown told BBC Newsnight: "I think No 10, the Metropolitan Police, and Sue Gray should get round a table and work out a way that this report can be published in full, unredacted, so that not only MPs, but our constituents more importantly, can make a judgement on what has happened."
Labour has called for the report to be published in full and the investigations to be wrapped up as soon as possible.
The SNP and Liberal Democrats are claiming the delay in publication is a "stitch-up" aimed at keeping Boris Johnson in power.
If anything less than that full report emerges next week, we can expect even more condemnation from the opposition parties.
All this is crucial to the prime minister's political future because some Conservative MPs have been telling me that they're considering putting in a letter of no confidence once they've read Sue Gray's report.
If that report is shorn of all the serious allegations of what was going on in Downing Street, it may well be the case that they sit on their hands until the Met completes its work.
The opposition say this is obviously to the advantage of Boris Johnson - it buys him some time.
But there's a bigger issue too, because it's not just the opposition politicians.
Some Conservatives are saying to me too that the way all this has been handled has further eroded public trust in the political system.
Mr Johnson has been under pressure following a string of allegations about events held in Downing Street and other government premises during coronavirus restrictions.
The SNP's Westminster leader, Ian Blackford, said: "If the UK government refuses to publish the full unredacted report it will prove, yet again, that Westminster is utterly corrupt and broken beyond repair."
Liberal Democrat leader Sir Ed Davey said: "Anything short of the full report would be a Whitehall whitewash not worth the paper it is written on."
And Labour leader Sir Keir Starmer said: "What I want to see is Sue Gray's report in full and the investigation finished as quickly as possible."
Meanwhile, senior Conservative MP Tom Tugendhat has become the first MP to publicly say he would consider running for Tory leader if a contest was triggered.
Mr Tugendhat, who chairs Parliament's foreign affairs select committee, told Times Radio "it's up to all of us to put ourselves forward" but he added there was "not a vacancy at the moment" and said he had not been canvassing support.