Rishi Sunak is to consult his ethics adviser later about Suella Braverman’s handling of a speeding offence.
The home secretary sought advice via civil servants and an aide about arranging a private speed awareness course while attorney general in 2022.
A government source has denied her actions broke the ministerial code.
Labour’s deputy leader Angela Rayner said the prime minister should order his adviser to investigate whether rules were breached “without delay”.
The Liberal Democrats are also calling for an investigation and said Mr Sunak needed to make a statement in Parliament about the claims.
Meanwhile, Mrs Braverman is due in the Commons on Monday afternoon for Home Office questions.
She was caught speeding last summer, and faced three points on her licence and a fine, or a course as part of a group.
The home secretary is under scrutiny, not over the speeding offence itself, but over whether she acted properly in trying to arrange a one-to-one awareness course.
In a letter to Mr Sunak, Ms Rayner said: “Members of the cabinet are subject to the same laws as the rest of us, and any attempt to direct civil servants to obtain special treatment in this matter would clearly amount to an unacceptable abuse of power and privilege by the home secretary.”
In a separate statement, Ms Rayner said the prime minister needed to “show some backbone” and order Sir Laurie Magnus, the independent adviser on ministerial interests, to “get to the bottom of this episode”.
The prime minister, who has been in Japan for the G7 summit, will speak to Sir Laurie on his return. He cannot begin an investigation into a minister without Mr Sunak signing-off on an inquiry.
After being caught speeding, Mrs Braverman was offered the choice of either a fine and points on her driving licence, or a speed awareness course.
A government source told the BBC the senior minister had been “concerned” about her insurance premiums, and favoured doing a course.
She asked civil servants about arranging a course for just her, citing security concerns about doing one as part of a group, but was told it was not a matter for the civil service.
Mrs Braverman then asked a special adviser to try to arrange a private course.
When the course provider told her there was no option to do this – and after she was reappointed home secretary in Mr Sunak’s government – she opted to pay the fine and accept the points because she was “very busy” and did not have the time to do a course, the BBC has been told.
The same government source refused to say whether Mrs Braverman’s motivation to do the course in private was to reduce the chances of her being recognised by members of the public.
The ministerial code sets standards of conduct expected of ministers, including that they must uphold the political impartiality of the civil service.
Speaking to the Westminster Hour on BBC Radio 4, former senior civil servant Sir Philip Rycroft said Mrs Braverman’s reported actions appeared to be a “real lapse of judgement”.
“Obviously, there’s still investigations to be done and so on but the code is very clear. Ministers must ensure that no conflict arises or appears to arise between their public duties and their private interests.
“Even asking a question of a civil servant as to how she might go on one of these courses puts them in an impossible position.”
Mr Sunak apparently did not know anything about the story the until it was first reported in the Sunday Times. and he declined to say whether he would be ordering an investigation, when asked at the G7 summit.
Speaking at a news conference, he also declined to say he backed Mrs Braverman – but a Downing Street source later said that “of course” he did.
“I don’t know the full details of what has happened, nor have I spoken to the home secretary,” Mr Sunak said.
“But I understand she has expressed regret for speeding, accepted the penalty and paid the fine.”
After serving as attorney general between February 2020 and September 2022, Mrs Braverman was promoted to home secretary under Liz Truss.
She resigned on October 19 after sending an official document from a personal email to a backbench MP – describing it as a “technical infringement of the rules”. But she was reappointed to the same role by Mr Sunak six days later following the collapse of the Truss government.
A source close to the home secretary said: “Mrs Braverman accepted three points for a speeding offence which took place last summer.
“The Cabinet Office was made aware of the situation as requested by Mrs Braverman. She was not and is not disqualified from driving.”
A Cabinet Office spokesman said: “It would not be appropriate to comment on the existence or content of advice between government departments.”