Mr Raab, who is also Justice Secretary, is facing two separate claims of bullying officials when he held previous government roles.
He has denied the allegations, insisting he would “thoroughly rebut and refute” the claims, but last week asked the PM to launch an inquiry into his own conduct.
Now Mr Sunak has asked Mr Tolley, a King’s Counsel at Fountain Court Chambers, to conduct the inquiry.
If Mr Raab is found to have broken the ministerial code, then it could force him to resign. It could also raise fresh questions over the Prime Minister’s judgment following the resignation of Cabinet Office Minister Sir Gavin Williamson earlier this month following complaints he sent text messages berating former Chief Whip Wendy Morton.
According to the Fountain Chambers website, Mr Tolley’s practice covers commercial litigation in banking, financial services and professional negligence.
The website adds that he also covers employment tribunal litigation specialising in “issues of post-termination restrictions and confidentiality” and “equal pay, discrimination and whistleblowing claims”
In 2001 he represented King Charles III while he was the Prince of Wales, in an employment tribunal case involving a former secretary who claimed she had been racially abused and forced out of her job. Elizabeth Burgess lost her claim of constructive dismissal after the tribunal ruled the former secretary could not prove any of the allegations.
The Prime Minister’s spokesman said Mr Tolley had been asked to look into the two formal complaints made against Mr Raab but that he could discuss widening the scope of his investigation with Mr Sunak if other issues arise.
The spokesman said: “The Prime Minister has appointed Adam Tolley to conduct the investigation, establish the facts and find his findings.”
Although the spokesman stressed that Mr Tolley’s investigation was independent and he would be able to access all the information “he wishes to see”, the Prime Minister will take the final decision on any disciplinary action against Mr Raab.
Mr Sunak is also looking to appoint a new Independent Ethics Adviser to replace Lord Geidt who quit in June conceding the prime minister may have broken the ministerial code over the Partygate scandal.
The PM’s spokesman made it clear that Mr Tolley’s appointment to conduct the inquiry into Mr Raab was entirely separate to that role.
He said: “He [Mr Tolley] will be providing advice on a specific issue rather than the independent adviser who provides advice on the ministerial code. So he is establishing the facts on a specific incident rather than making judgments across the piste on the code.”
The spokesman added that the final report will be published in a “timely manner” but that no specific timeframe would be announced.