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 leader Sir Keir Starmer said the prime minister should order his adviser, Sir Laurie Magnus, to investigate whether rules were broken.
Speaking to BBC Breakfast, Sir Keir said it looked like “inappropriate action took place” from the home secretary that “needs to be fully investigated”.
“The usual consequence of breaking the ministerial code is that you’ll go,” he added.
Labour’s deputy leader Angela Rayner also called on the prime minister to “show some backbone” and order Sir Laurie to “get to the bottom of this episode”.
Mr Sunak, who has been in Japan for the G7 summit, will speak to his ethics adviser on his return. Sir Laurie cannot begin an investigation into a minister until the prime minister gives it the go-ahead.
The Liberal Democrats are also calling for an investigation and said Mr Sunak needed to make a statement in Parliament about the claims.
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.
The ministerial code sets standards of conduct expected of ministers, including that they must uphold the political impartiality of the civil service.