Raab said in a letter to Sunak that he had been notified of two separate complaints, one from when he was foreign minister and one from his tenure as justice minister.
Some officials who have worked for him say he was a tough and demanding boss, with others being quoted by local media as saying he was rude, aggressive and a bully.
Raab has denied the allegations.
Raab said he was notified of the complaints this morning, both relating to his behaviour when he served under former prime minister, Boris Johnson.
“I am confident I behaved professionally throughout but, of course, I will engage thoroughly and look forward to transparently addressing any claims,” he said.
The new allegations mount pressure on Sunak, who used his first speech as prime minister last month to say his government would demonstrate “integrity, professionalism and accountability”.
Sunak defended his deputy this week, saying he did not recognise allegations that Raab had bullied staff.
Cabinet minister Gavin Williamson resigned from government last week over allegations he had bullied colleagues, with opposition parties saying the case raised questions about Sunak’s judgment just weeks into the job.
In response to Raab’s letter, Sunak said he knew his deputy would be “keen to address the complaints”, adding that it was “right that these matters are investigated fully”.
Opposition Labour Party politicians wore anti-bullying badges in parliament today, when Raab stood in for Sunak at the weekly prime minister’s questions because he was returning from a G20 meeting in Indonesia.
Labour’s deputy leader, Angela Rayner, asked Raab to apologise for his behaviour because staff were too scared to enter his office and accused Sunak of being “too weak” to deal with problematic cabinet ministers.
Asked if he had ever had requested a non-disclosure agreement related to his behaviour, Raab said no, but there was an employment dispute before he entered parliament and this included a confidentiality clause, which he said was standard at the time.
Raab, who stood in for Johnson for almost a month in 2020 while the then-prime minister was hospitalised with Covid-19, said he sought to set “high standards” in his teams to deliver in challenging circumstances over recent years.
A spokesman for the prime minister said Sunak had confidence in Raab and was working to appoint an independent investigator to look into the complaints.