Gayle Smith, chief executive of One, said it launched an internal investigation after ex-employees at the anti-poverty charity's Johannesburg office aired grievances in November.
In a blog on the charity's website, Smith said: "The investigation yielded evidence of unprofessional conduct and, in particular, what I would characterise as bullying and belittling of staff between late 2011 and 2015 in our Johannesburg office.
"Staff were called names, and some said their manager put them to work on domestic tasks in her home.
"The investigation also found the situation was not adequately addressed nor resolved by executive management at the time and One's board was not, in my view, properly or fully informed."
Smith added that one former employee who met One staff and a mediator to discuss her complaints described how her manager had made "sexist and suggestive comments about her to a government official from another country".
Smith said that when some former employees informed the charity last week of legal action, a fresh claim emerged that the member of staff had been demoted because she refused to "become intimate" with the official.
"We have not been able to corroborate these appalling claims," said Smith. "We do not discount any allegation – we investigate them and will continue to do so should others arise."
According to the Mail on Sunday newspaper, which said it had uncovered a "toxic culture of bullying and abuse", the charity faces compensation claims worth £3.6m.
Smith said the charity "needed to own an institutional failure" and had voluntarily shared its investigation's findings with the commission.
The One Campaign is registered as a not-for-profit corporation in the United States.
One Against Poverty – the full name of the associated UK charity – was registered with the commission in 2016, after the reported incidents took place.
A commission spokeswoman said: "We can confirm that the registered charity, One Against Poverty, has submitted a serious incident report to the commission regarding historic allegations of bullying at the Johannesburg branch of the One Campaign.
"Although these are separate organisations, we encourage charities to consider and report on any issues surrounding their partner organisations because these might affect public trust and confidence in their charity.
"The One Campaign does not fall under our regulatory remit. However, we will be engaging with the trustees of One Against Poverty to ensure that its staff are properly protected and that a commitment to safeguarding is fully embedded across the charity’s culture."
Bono, who co-founded One in 2004, said in a statement to the Mail: "We are all deeply sorry. I hate bullying; can't stand it.
"The poorest people in the poorest places being bullied by their circumstances is the reason we set up One.
"So to discover last November that there were serious and multiple allegations of bullying in our office in Johannesburg left me and the One board reeling and furious. You question the whole reason you're doing this."
The U2 singer praised Smith for taking "swift and decisive action to address what had gone badly wrong".
He added: "Although the bullying allegations centre on an individual (an accomplished female executive formerly of the African Development Bank and the World Bank), the head office failed to protect those employees and I need to take some responsibility for that.
"In fact, if they would agree, I would like to meet them and apologise in person."