According to The Times, Sue Gray has not found sufficient evidence to refer it to the police.
Only the prime minister can have the power to bring about an investigation into whether the ministerial code has been breached.
Gray's inquiry into the reports of multiple parties being held over lockdown has been repeatedly cited as a reason as to why government ministers cannot answer questions about whether the gatherings took place and if they breached lockdown rules in place at the time.
It comes amid the most recent reports of a No 10 party being held the night before the Duke of Edinburgh’s funeral.
The inquiry has been criticised in some quarters amid questions of its independence. As a civil servant, Sue Gray ultimately reports into the prime minister.
Labour MP Jess Philips told BBC's Question Time: “Sue Gray is not independent. It doesn’t matter how independently minded she is. She’s a government civil servant.
Read more: Boris Johnson’s former adviser apologises for No10 party night before Prince Philip’s funeral
“It’s like me asking a member of my staff to look into me and then report to me before reporting to parliament about what I have done wrong.
“The reality is she is not independent of Boris Johnson.”
And Hannah White, deputy director of the Institute for Government, said: “Sue Gray’s report is internal, NOT independent – her boss is a Cabinet minister.
“If they wanted an independent report they’d have brought in a judge or similar.”
The Met said on Thursday it would not be investigating the matter itself.
However, after making that statement reports of two further parties held at Number 10 were made on the night before the Duke of Edinburgh's funeral, when the Queen was forced to sit alone to abide by social distancing rules.
Read more: Renewed calls for PM to resign after reports Number 10 staff partied on eve of Prince Philip's funeral
According to The Telegraph, a leaving party was held for James Slack, who previously worked as Boris Johnson's director of communications.
The newspaper reported that the raucous party saw a staff member going to the shops with a suitcase to buy alcohol, and a swing used by the PM's toddler son was broken in the garden during the gathering.
Slack apologised for the event, adding: “This event should not have happened at the time that it did. I am deeply sorry, and take full responsibility.”
However, although Downing Street did not deny the details in the report, a spokesperson described the event as a “leaving speech”.
A Number 10 spokesperson said on Friday: “On this individual’s last day he gave a farewell speech to thank each team for the work they had done, both those who had to be in the office and on a screen for those working from home.”
Watch: Everything you need to know about Sue Gray's inquiry
The prime minister was not in Number 10 at the time and is believed to have been at Chequers.
These allegations are the latest in a long line of accusations government officials held parties inside Downing Street while the rest of the country was banned from seeing friends and family.
On Wednesday, Johnson issued a grovelling apology after he admitted attending a party in the gardens of Downing Street during the height of the first lockdown, when social gatherings were banned under COVID legislation.
He is not making public appearances this week after a family member tested positive for COVID on Thursday morning.
Despite a number Cabinet ministers rallying to defend him, Johnson has been facing increasing calls to resign as prime minister.Internet Explorer Channel Network