Dominic Raab has said he is “confident” he has behaved “professionally”, as the deputy prime minister faces an investigation into two formal complaints against him.

The Justice Secretary confirmed on Wednesday that two separate complaints had been made about his conduct, as the Prime Minister agreed to open an independent investigation into the allegations.

Rishi Sunak still has full confidence in Mr Raab, Downing Street said, with the prime minister set to appoint an “independent” investigator to examine the complaints made against Mr Raab, in the absence of a permanent ministerial ethics watchdog.

But a Downing Street spokeswoman said Mr Sunak will not be obliged to accept the findings of any report produced by the investigator as the Prime Minister remains the “ultimate arbiter” of the ministerial code.

The lack of an ethics adviser had raised immediate questions about how an independent investigation into Mr Raab’s conduct would be carried out, with the Justice Secretary pressed by Labour’s Angela Rayner at Prime Minister’s Questions about when such an official would be appointed.

Mr Raab, standing in at Prime Minister’s Questions for Mr Sunak who is flying back from the G20 summit in Bali, faced an onslaught of questions from Labour’s deputy leader as he said he was looking forward to “transparently addressing any claims that have been made”.

Mr Raab has been facing a series of allegations he bullied officials and deployed rude and demeaning behaviour in previous Cabinet roles.

In a letter to Mr Raab on Wednesday, Mr Sunak told his deputy that “integrity, professionalism and accountability are core values of this Government” and said that an investigation was the “right course of action”.

But Labour rounded on the absent prime minister in the Commons, with Ms Rayner telling MPs that Mr Raab “has had to demand an investigation into himself, because the prime minister is too weak to get a grip”.

“The deputy prime minister knows his behaviour is unacceptable, so what is he still doing here?”

Ms Rayner told the Commons: “After days of dodging and denial, this morning the Deputy Prime Minister finally acknowledged formal complaints about his misconduct, but his letter contains no hint of admission or apology.

“This is anti-bullying week. Will he apologise?”

Mr Raab said: “She asks about the complaints, I received notification this morning, I immediately asked the Prime Minister to set up an independent inquiry into them.

“I’m confident I behaved professionally throughout but of course I will engage thoroughly and look forward to transparently addressing any claims that have been made.”

In his letter to the Prime Minister, Mr Raab said: “I have just been notified that two separate complaints have formally been made against me, in parallel, from my time as foreign secretary and my first tenure as justice secretary, which ended in September of this year.

“I am, therefore, writing to request that you commission an independent investigation into the claims as soon as possible. I will co-operate fully and respect whatever outcome you decide.”

The Conservative MP for Esher and Walton told Mr Sunak he had “never tolerated bullying, and always sought to reinforce and empower the teams of civil servants working in my respective departments”.

Downing Street said that work has begun to appoint the investigator, who will be someone from outside the Government, but would not say when the person would be in post or when the inquiry would be completed.

Asked how the process could be considered independent with an investigator hand-picked by the Prime Minister, a spokeswoman said: “It will be a suitably qualified, independent person to investigate the complaints.”

Mr Sunak also wants to appoint a permanent independent adviser on ministerial interests “as quickly as possible”, Downing Street said, to fill the post that has been vacant since Lord Geidt quit in June.

Labour hit out at the decision on Wednesday afternoon, with Ms Rayner warning against a “Tory whitewash”.

Calling on Mr Sunak to provide assurances that the investigator’s findings will be “transparent and acted upon”, she said: “A genuinely independent investigation into Dominic Raab is needed to uphold the Ministerial Code and shed light on these serious bullying allegations.”

Top Ministry of Justice officials had reportedly ruled there must be a senior civil servant in the room at all meetings involving Mr Raab due to the recent allegations, according to The Guardian on Wednesday.

The newspaper also reported that Philip Rycroft, the former permanent secretary to the Department for Exiting the European Union, raised concerns about Mr Raab’s behaviour during his time as Brexit secretary with the then-cabinet secretary Mark Sedwill.

Mr Raab was also reportedly warned about his behaviour towards officials while he was foreign secretary.

The concerns were raised with Mr Raab by Lord Simon McDonald, who was the senior civil servant at the Foreign Office, and the mandarin also informally discussed the situation with the Cabinet Office’s propriety and ethics team, The Guardian reported.

In a radio interview on Monday, Lord McDonald had acknowledged that allegations Mr Raab could be a bully were plausible.

It is the latest blow to the new prime minister’s administration, after he faced criticism for appointing Sir Gavin Williamson to his senior team despite being told he was under investigation for allegedly bullying a colleague, claims that caused Sir Gavin to quit.

Simon Clarke, the former levelling up secretary, was among those defending Mr Raab on Wednesday.

Speaking on BBC Radio 4’s PM programme, he said Mr Raab was a “hard taskmaster” but was not a “bully”.

The Tory MP said: “I know Dominic very well and, indeed, I worked for him for three years, and I can say this, he’s a hard taskmaster. He works hard, he asks a lot of himself as well as all those around him. But he is not a bully.

“He’s a good person …. someone who is always looking to demand the best of himself and others.”