The probe into two formal complaints against the justice secretary is finally under way after Mr Sunak appointed prominent employment lawyer Adam Tolley KC as lead investigator.
Downing Street said Mr Tolley would be given access to witnesses and documents such as emails and WhatsApp messages, and his report would be published in full after its completion.
But while the investigator can make recommendations about sanctions to the prime minister, Mr Sunak himself holds the power to decide whether Mr Raab can stay in the government.
And the barrister will need to obtain the PM’s permission if he wishes to extend his probe beyond the complaints already submitted about Mr Raab’s alleged behaviour during an earlier stint at the Ministry of Justice and his time as foreign secretary.
Labour deputy leader Angela Rayner called for Mr Tolley to be given free rein to consider any allegations against Mr Raab, amid media reports that multiple civil servants are considering complaints about aggressive behaviour.
Ms Rayner called for him to look into claims that the MoJ’s top civil servant, permanent secretary Antonia Romeo, called officials at the Home Office to apologise for the way Mr Raab had acted in a meeting.
Ms Rayner warned of the danger that the inquiry would prove to be a “whitewash”, accusing the Conservative government of “a troubling track record of brushing serious misconduct under the carpet”.
And Liberal Democrat chief whip Wendy Chamberlain called on Mr Sunak to commit now to sacking Raab if the bullying complaints were upheld.
“Anything less would make a mockery of his promise to bring back integrity,” said Ms Chamberlain.
Mr Tolley has a commercial and employment practice at Fountain Court Chambers in London and has represented a string of establishment clients in cases involving allegations of harassment and discrimination at work.
In 2012, he appeared for an unnamed security service defending sexual harassment allegations brought by a female secret agent who claimed she was “blacklisted” after complaining that she was being pestered by her boss.
He represented King Charles in a number of claims against his household as Prince of Wales, including the successful defence of a case in 2005 in which a former secretary alleged sexual discrimination and unfair dismissal. He has also appeared at employment appeal tribunals for the Ministry of Defence and the Department for Environment, Food and Rural Affairs.
The justice secretary asked Mr Sunak to order an inquiry after the two formal complaints were submitted on 15 November, and has vowed to “thoroughly rebut and refute” allegations against him.
But the launch was delayed for a week while an investigator was recruited.
Normally, a probe of this kind would be carried out by Downing Street’s independent adviser on ministerial ethics. But the post has been empty since Lord Geidt quit in June in protest at Boris Johnson’s apparent readiness to breach the law.
Mr Sunak has promised to appoint an independent adviser, but no candidate has yet been named.
No timetable has been set for Mr Tolley’s inquiry, but it is thought likely that he will report by Christmas. Downing Street said that his report would be released “in a timely way” but would not commit to immediate publication.
Terms of reference released by the Cabinet Office say that information provided to Mr Tolley will be received “in confidence”. His investigation should be completed “as swiftly as possible”.
The KC will provide a narrative of events surrounding the alleged bullying and may provide “advice” to Mr Sunak on how he should respond in relation to potential breaches of the ministerial code of conduct or employment law, said No 10.
But as final arbiter of the code, it will be for Mr Sunak himself to decide whether his deputy should be sacked or face any lesser punishment, such as being required to issue an apology.
Ms Rayner said: “There must be no hint of a whitewash when it comes to the slew of serious allegations the deputy prime minister now faces.
“The scope of this investigation must immediately be expanded to enable proactive investigation of Dominic Raab’s behaviour during his time as a minister, including so-called expressions of concern, informal complaints and the concerning testimony of his own former permanent secretary.
“A temporary stop-gap investigator, appointed in a panic, with an absurdly narrow remit is not a solution to dealing with the flood of allegations of ministerial misconduct now requiring investigation.
“Rishi Sunak pledged integrity and accountability, but his broken promise to appoint an ethics watchdog as one of his first acts as prime minister shows that he is already failing to stop the rot in Downing Street.”
Ms Chamberlain said: “It’s only been four weeks, and already Rishi Sunak has repeatedly turned a blind eye to allegations of inappropriate behaviour by Conservative ministers. This can’t be yet another case of one rule for Conservative MPs and another rule for everyone else.
“Every day brings yet more sleaze and scandal, while families around the country pay the price for this endless Conservative chaos.”