It is understood Mr Johnson and Sue Gray, a senior civil servant, met at least once for an update on the document's progress, but a Whitehall source said that its contents were not discussed at any point.
Both sides admit a meeting took place but both also deny instigating it, British media reported on Saturday.
For Mr Johnson, the pressure is coming from opposition MPs who are demanding transparency and details of the “secret meeting”.
He was issued with a fixed penalty notice for breaking Covid rules but the ‘partygate’ scandal has spread to numerous government departments holding lockdown-busting events.
About 30 people, including Mr Johnson, are being contacted by the Cabinet Office to warn them of the contents of the document before its publication.
Angela Rayner, the opposition Labour's deputy leader, called on the prime minister to “urgently explain” why the “secret meeting” had taken place.
She said public confidence in the process had already been “depleted”, and people “deserve to know the truth”.
“Boris Johnson must urgently explain why he held a secret meeting with Sue Gray to discuss her report despite claiming her investigation was completely independent.
“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.”
Liberal Democrat MP Christine Jardine also called on the PM to throw light on the meeting.
“Any whiff of a stitch-up would make an absolute mockery of the report,” she said.
“This meeting must be explained. Most of the country already feels betrayed by Boris Johnson. He has undermined public trust and the public deserve a process that is completely transparent and beyond question.
“It is a sorry state of affairs that Boris Johnson and his government are in such a crisis, he should do the right thing and resign, if not Conservative MPs should sack him.”
His Downing Street office said Mr Johnson acted in a legitimate and transparent manner.
“The prime minister commissioned the investigation led by Sue Gray and has been clear throughout that it should be completely independent," a No 10 representative said.