It is thought that most of the letters containing the warnings were sent out on Thursday, as the Metropolitan Police concluded their investigation into lockdown breaches in Downing Street and across Whitehall.
Mr Johnson is set to be among those approached by the Cabinet Office team but No 10 had no update as to whether he had received a letter on Friday evening.
Not everyone contacted will necessarily be named or identified in the report — it may merely be that their actions are referenced because they are pertinent to the narrative of a particular event.
The investigators are understood not to be including the full details of the contents in the letters but instead are giving a “gist” of the relevant sections to those being approached.
Earlier, Mr Johnson insisted he would not seek to block names appearing in the senior civil servant’s report and said he was “looking forward” to seeing its contents “pretty soon”.
The prime minister said he was “very grateful” for the Met’s investigation, which concluded with the issuance 126 fines.
There have been demands for answers over how Mr Johnson escaped with only one fixed penalty notice despite being believed to have been present at a number of rule-breaking events.
Potentially damaging further details, including the names of some of those who attended gatherings, will be published in the report.
Sources close to the investigation expect it will be published early next week.
Mr Johnson declined to apologise again for the rule-breaking in No 10 during a visit to Hilltop Honey’s factory in Newtown in Powys, Wales, on Friday.
Asked if No 10 would be blocking any names from appearing in the report, he said: “That will be entirely up to Sue Gray and I’ll be looking forward very much to seeing what she has to say, and fingers crossed, that will be pretty soon next week.”
Officials affected by the Gray inquiry, including those to be named, are being contacted so they can have a chance to respond before the current draft is finalised for publication.
There were calls for Scotland Yard to explain how Mr Johnson had received only the one fine for his 56th birthday gathering in June 2020 when indoor mixing was banned.
Former director of public prosecutions Lord Ken Macdonald said: “It’s very difficult for us to disentangle exactly how the police investigation has proceeded and how fair it’s been.”
He told BBC Radio 4’s Today programme: “I think without the police providing an explanation for that it’s very difficult for us to understand why they came to the conclusions that they did.”
There has also been no explanation as to why Cabinet Secretary Simon Case, who had to recuse himself from running the civil service inquiry after reports of a Cabinet Office Christmas party surfaced, has not been fined.
Mr Johnson, his wife, Carrie Johnson, and Chancellor Rishi Sunak were all fined over the birthday gathering.
But both Johnsons were told by police they would face no further action, Downing Street said, and Mr Sunak has not received an additional fine.
The Met said 28 people had received between two and five fines.