Boris Johnson is coming under pressure to declare when he paid for Downing Street flat refurbishments.
Labour has urged the PM to appoint a fully “independent” figure to oversee ministers’ conduct, including his own.
Lord Geidt – announced on Wednesday as Boris Johnson’s new standards adviser – will look into the funding of the recent Downing Street flat renovations.
Labour leader Sir Keir Starmer said it was wrong that Lord Geidt needed to get the PM’s permission before carrying out this and other inquiries.
But No 10 said “successive” advisers had been taken on under these terms.
The prime minister receives an annual public grant of £30,000 to carry out renovations to the Downing Street flat each year.
But newspaper reports suggest the bill for the latest work could be as high as £200,000.
Mr Johnson has said he paid the costs himself, but he has not specified whether this happened when he first received the bill, or whether he was loaned the money and later repaid it.
Normally, MPs have to register within 28 days any donations or loans which could influence their actions – and there is a list of ministers’ interests with separate reporting rules.
The government insists the prime minister “acted in accordance with the appropriate codes of conduct and electoral law”.
Meanwhile, the Electoral Commission – the watchdog overseeing political finances – has launched an investigation which will assess the Conservative Party’s compliance with laws on political donations.
It said there were “reasonable grounds to suspect that an offence or offences may have occurred”.
On Wednesday, it was announced that Lord Geidt, a crossbench – non-party-affiliated – peer, would take over as Mr Johnson’s adviser on the ministerial code – the rulebook for conduct in public office.
Downing Street also said he would carry out his own inquiry into the funding of the renovations and “advise the prime minister on any further registration of interests that may be needed”.
However, the prime minister’s spokesman said Mr Johnson remained “the ultimate arbiter” of the ministerial code and would decide whether to accept or reject any findings.
Sir Keir told ITV’s Peston show the investigations into the funding of the Downing Street flat renovation “could be over in five minutes if the prime minister just answered the question: Who paid for it in the first place?”.
On dealing with the ministerial code, he said: “I’d have an independent adjudicator.
“The idea that an inquiry can’t get going unless the prime minister says it can get going, and it can’t come to conclusions that are binding unless the prime minister says so… undermines the whole fabric.
“I would bring in changes, an independent adviser who can actually start investigations of his or her own volition.”
The FDA senior civil servants’ union is currently taking legal against the government over its stance on the ministerial code.
It argues that decisions made under it should be subject to appeal in the courts – but the government says the code is separate from the law and should remain so.
FDA general secretary Dave Penman told the BBC: “The prime minister has ensured that not only is he the final arbiter of any investigation, but he retains the veto on whether an investigation starts in the first place.”
He added: “Few will take comfort from his retention of almost complete control of the process.”
But Downing Street said that, before becoming the prime minister’s adviser, Lord Geidt had agreed new terms of reference, allowing him to “raise issues of concern to the attention of the PM on a confidential basis”.
He would also be able to “provide recommendations on sanctions following an investigation, enhancing the advice available to the prime minister”, a statement added.
The adviser role had “always been a direct appointment of the prime minister under successive administrations”, the statement said, adding: “The same process has been followed here.”