An investigation into Deputy Prime Minister Dominic Raab’s conduct could range more widely than just the two formal complaints made against him.

Mr Raab requested the independent investigation into the complaints about his behaviour while he was foreign secretary and during his first period in the Ministry of Justice.

Downing Street suggested it would be open to the individual appointed to carry out the inquiry whether they want to look at other allegations made about the Justice Secretary’s conduct.

Number 10 has said a “suitably qualified, independent person” will investigate the allegations about bullying behaviour, although no appointment has yet been made.

The latest allegations against Mr Raab suggest he behaved so badly in a meeting with the Home Office during his first stint as justice secretary that his department’s top official had to personally apologise to counterparts afterwards.

The Guardian reported he acted “so badly and inappropriately” at a high-level meeting that the permanent secretary at the Ministry of Justice, Antonia Romeo, was forced to call senior officials of then home secretary Priti Patel to express regret.

The Prime Minister’s official spokesman said: “The Prime Minister has asked for an independent investigator to establish the facts and provide their findings.

“In the first instance, we will appoint an investigator and then there will be the process.”

The spokesman said “it will be for the investigator to decide what they do or do not choose to look at”, indicating that meant they could go beyond the two formal complaints.

“They will have a remit. I imagine we will be able to give an overview once an appointment is confirmed of what they will look into,” the spokesman said.

“Obviously they will be able to look into issues as they see fit.”

Mr Raab wrote to Rishi Sunak last week to request the investigation into the two formal complaints.

The investigator will be appointed by the Prime Minister and report to him, although Mr Sunak will not be obliged to accept any findings made about Mr Raab’s conduct.