Mr Sunak is being investigated over whether a declaration of interest was "open and frank", under rules set out by the commissioner for standards.

The BBC understands the probe relates to a childcare firm his wife has shares in.

The commissioner decides whether an MP has broken rules after an inquiry.

A Downing Street spokesperson said: "We are happy to assist the commissioner to clarify how this has been transparently declared as a ministerial interest."

Last month, Mr Sunak faced questions over shares his wife, Akshata Murty, holds in Koru Kids, a childcare agency that could benefit from a new policy unveiled in the spring Budget.

Chancellor Jeremy Hunt announced a pilot of payments for new childminders, with more for those who sign up through agencies.

Ms Murty was listed as a shareholder in one of those agencies, Koru Kids, as recently as 6 March.

Mr Sunak did not mention Ms Murty's links to Koru Kids when he was questioned by MPs over the childcare policy at a parliamentary committee hearing on 28 March.

Labour MP Catherine McKinnell asked Mr Sunak whether he had any interest to declare, and in reply he said: "No, all my disclosures are declared in the normal way."

In a letter to the committee, sent a few days after the hearing, Mr Sunak said his wife's interest was declared to the Cabinet Office and that an updated statement of ministers' interests would be due out shortly.

In his letter, Mr Sunak said the the list of ministerial interests "ensures steps are taken to avoid or mitigate any potential conflict of interest".

The list of ministerial interests is separate to the register of interests for MPs, which says members "must always consider whether they have a conflict of interest".

The list has not been updated for nearly a year and was last compiled by Lord Geidt, who resigned as Boris Johnson's ethics adviser.

The prime minister's spokesman said that the commissioner was investigating whether Mr Sunak was not clear that he had dual obligations to declare the connection as a ministerial interest and also to declare it when speaking to MPs about the issue.

"The prime minister has set out in his response to the Liaison Committee that he is confident the appropriate process has been followed to avoid or mitigate any potential conflict of interest, and that the interest of ministers' spouses or partners is not something that would influence their actions either as ministers or as members of parliament," he said.

Labour's deputy leader Angela Rayner MP said the failure to update the rules or publish the register of ministers' interests had "left a transparency black hole which is enabling the prime minister and those he has appointed to dodge proper scrutiny of their affairs".

She added: "If Rishi Sunak has got nothing to hide, he should commit to publishing the register before May's elections so the public can see for themselves."

The Liberal Democrats said the investigation was another example of a Conservative prime minister allegedly "bending the rules".

"After months of Conservative sleaze and scandal, the public just want a government which is focused on the country, rather than saving their own skin," Liberal Democrat chief whip Wendy Chamberlain said.

Code of conduct

An update on the commissioner's website says Mr Sunak is being investigated under paragraph 6 of the code of conduct for MPs.

The paragraph reads: "Members must always be open and frank in declaring any relevant interest in any proceeding of the House or its committees, and in any communications with ministers, members, public officials or public office holders."

The commissioner for standards is an independent officer who investigates allegations that MPs have breached Parliament's code of conduct.

Following investigation, if the watchdog thinks the allegation represents a breach of the code, they can put such cases before MPs sitting on the Committee on Standards, who can decide any sanctions.

Breaching the rules on standards can lead to serious consequences for some MPs, including suspension from the House of Commons. There are many of these investigations every year but most end with a minor telling off from the commissioner.

Mr Johnson has been a serial offender when it comes to the late declaration of earnings and Labour leader Sir Keir Starmer had to apologise for something similar last year.

The rules are there for a reason though and it is all about transparency. MPs have a duty to be open about their financial affairs so colleagues and the public can assess whether there is any conflict of interest.

The whole system is undermined if MPs fail to follow the rules for whatever reason and most would expect a prime minister to be particularly careful about the paperwork.

The rules around lobbying in Parliament were tightened up in an updated version of the code of conduct, which was published in February following the controversy over paid advocacy work undertaken by former MP Owen Paterson.

New incentives

The pilot of bonuses for childminders was announced in the Budget on 15 March as part of the government's overhaul of childcare.

Mr Hunt said the government would be "piloting incentive payments of £600 for childminders who sign up to the profession, rising to £1,200 for those who join through an agency".

The pilot could drive up the number of childminders entering the profession and generate more business for companies such as Koru Kids.

Koru Kids is listed as one of six childminder agencies on the government's website.

On its website, Koru Kids welcomed the government's reforms and said "the new incentives open to childminders are great".

The website says new childminders would get a bonus of £1,200 if they "come through an agency like Koru Kids who offer community, training and ongoing support".