It is understood the pair held at least one meeting to give an update on the document's progress while it was being drafted but a Whitehall source said that its contents were not discussed at any point.
Such meetings would not have been viewed as unusual, the source said, with the aim to take stock of what stage the report was at.
It comes as around 30 people, including Mr Johnson, are being contacted by the Cabinet Office to warn them of the contents of the document ahead of its publication.
She said public confidence in the process had already been "depleted", and people "deserve to know the truth".
Downing Street insisted that the Prime Minister had been "clear throughout" that the report should be "completely independent".
It is thought that most of the Cabinet Office letters were sent out on Thursday, as the Metropolitan Police concluded its investigation into lockdown breaches in Downing Street and across Whitehall.
Mr Johnson is set to be among those approached but No 10 had no update as to whether he had received a letter on Friday evening.
"Public confidence in the process is already depleted, and people deserve to know the truth.
"This is a Prime Minister incapable of taking responsibility for the rotten culture he has created in Downing Street or of doing the decent thing.
"The Sue Gray report must be published in full and with all accompanying evidence."
A No 10 spokesperson said: "The Prime Minister commissioned the investigation led by Sue Gray and has been clear throughout that it should be completely independent.
"the decision on what and when to publish rests entirely with the investigation team and he will respond in Parliament once it concludes."
Sources close to the inquiry expect the report 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 Downing Street 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."
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.
A team of 12 detectives examined 345 documents, including emails, door logs, diary entries, witness statements and 204 questionnaires.
They also examined 510 photographs and CCTV images.