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

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

Standing in at Prime Minister's Questions for Rishi Sunak, who is flying back from the G20 summit in Bali, Mr Raab faced an onslaught of questions from Labour's deputy leader Angela Rayner 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 his response yesterday, Mr Sunak told the Justice Secretary that "integrity, professionalism and accountability are core values of this Government" and said that an investigation was the "right course of action".

Mr Raab came under pressure over his conduct, as 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."

Mr Sunak, who is in Indonesia at the G20 summit, continues to stand by Mr Raab but agreed to Mr Raab's proposal to hold an independent investigation.

The British Prime Minister wants to appoint a permanent independent adviser on ministerial interests "as quickly as possible", Downing Street said, to fill a post which has been vacant since Christopher Geidt quit in June.

But in the interim a new investigator will be appointed to look at the complaints made against Mr Raab.

A Downing Street spokeswoman said: "An independent investigator will be appointed by the Prime Minister to establish the facts and to provide their findings to him.

"This will be a prime ministerial appointment, they will report into him."

Work has begun to appoint the investigator, who will be someone from outside the Government, but No 10 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, the spokeswoman said: "It will be a suitably qualified, independent person to investigate the complaints."

Pressed on whether Mr Sunak would abide by the findings, the spokeswoman said: "The Prime Minister remains the ultimate arbiter of the (ministerial) code and the determination of any code issues arising from the findings will be a matter for the Prime Minister."

Labour also rounded on the new 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?"

Dominic Raab said he had been notified of two separate complaints

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".

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 yesterday.

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 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.

The Deputy Prime Minister, in his letter to Mr Sunak, said he had been "blessed to work with a wide range of outstanding civil servants, in particular my brilliant and dedicated private offices".

"I have always welcomed the mutual challenge that comes with serious policy-making and public service delivery.

"I have always sought to set high standards and forge teams that can deliver for the British people amidst the acute challenges that we have faced in recent years.

"I have never tolerated bullying, and always sought to reinforce and empower the teams of civil servants working in my respective departments."

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

Asked by LBC radio if characterisation of Mr Raab as someone who could bully and around whom bullying could happen, he replied: "Yes."

Mr McDonald added: "Dominic Raab is one of the most driven people I ever worked for, he was a tough boss.

"Maybe they are euphemisms, but I worked closely with him and I didn't see everything that happened."

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