Lord Geidt, the new adviser on ministerial standards, released his first report on ministers' interests on Friday, after months of delay.
He said Mr Johnson had not broken the ministerial code but might reasonably have been expected to be curious about the financial arrangements.
Lord Geidt also found there had been a "significant failing" from officials in how "rigorous" they were at examining the idea of setting up a trust to fund renovations to Downing Street.
The PM can use an annual public grant of up to £30,000 to decorate his Downing Street home, but it has been reported the renovations at No 11, where Mr Johnson lives with his fiancee Carrie Symonds, reached up to £200,000.
There had been discussions about a Downing Street Trust being set up to pay for the work, before legal advice received in June 2020 "raised doubts" about whether such a body "would be capable of dealing with costs associated with the private residences", said the adviser.
Mr Johnson's former chief aide, Dominic Cummings, claimed in April the Prime Minister had planned to have Tory donors cover the costs of the refurbishment, a move he deemed "unethical, foolish, [and] possibly illegal".
But Lord Geidt found Mr Johnson had not been aware that in lieu of any such trust to fund the refurbishment being set up, Tory donor Lord Brownlow had been involved in settling the bill.
His report said: "On October 20 2020, Lord Brownlow confirmed to Cabinet Office officials...that he had the day before settled an invoice for the No 11 Downing Street residence refurbishment works directly with the supplier.
"Cabinet Office officials appear not to have acted on this information to the extent of informing the Prime Minister.
"Moreover, despite the Prime Minister and Lord Brownlow having some limited contact during the following three months.”
Lord Geidt said "the Prime Minister - unwisely, in my view - allowed the refurbishment of the apartment at No 11 Downing Street to proceed without more rigorous regard for how this would be funded".
But he said the PM knew "nothing about" payments for the refurbishment work, which started while he was in hospital with coronavirus, until reports in the media surfaced.
He said: "I have also spoken ... to the Prime Minister who confirms that he knew nothing about such payments until immediately prior to media reports in February 2021.
"At that point, the Prime Minister immediately sought the necessary advice about his interests and, as a consequence, settled the full amount himself on March 8 2021."
Lord Geidt added: "It is clear from the record that while a serious and genuine endeavour, the (Downing Street) Trust was not subjected to a scheme of rigorous project management by officials.
"Given the level of the Prime Minister's expectations for the trust to deliver on the objects he had set, this was a significant failing."
A No 10 spokesman said: "Lord Geidt's independent report shows the Prime Minister acted in accordance with the ministerial code at all times.
"The Prime Minister has made a declaration in his List of Ministerial Interests, as advised by Lord Geidt.”