One former worker told BBC Two's Newsnight that staff were told of complaints about a client who sexually exploited girls he met there, off site.
But she says these concerns were not passed on and Newsnight has established that authorities were not alerted.
The charity said it would have told police of any allegation it knew about.
Two witnesses allege complaints were made known to staff at the charity's Urban Academy in Southwark, south-east London, which provides educational support to over-16s with "complex emotional or behavioural needs".
The charity's chairman, Alan Yentob, who is also the BBC's creative director, said: "There is no evidence that we were informed about a sexual abuse case. I think this amount of rumour and allegation and counter-allegation… is disgraceful."
Before it shut its doors on Wednesday, Kids Company provided practical, emotional and educational support to some of the most deprived and vulnerable inner-city children and young people in London, Liverpool and Bristol.
The charity, founded in 1996 by Camila Batmanghelidjh, was well known for its famous and wealthy supporters who included David Cameron, but has been closed amid a row over funding.
In addition to the allegations being made to Newsnight, a former client of the charity has told BBC News that, when she was 16, she was touched inappropriately by a 24-year-old man who was also a client of the charity at that time.
She said she attempted to tell a teacher at the charity, which provides educational services, but the conversation was stopped before she could explain she had been assaulted.
Kate (not her her real name) said the sexual assaults took place over a period of seven months in 2009, and occurred at one Kids Company site.
"He would touch me in inappropriate places," she said.
"He would have his dirty comments like how he would like to do certain things. I got pushed to the wall once as well when he threatened me and said that if I didn't comply to his wishes then he would get me."
Kate explained what happened when she tried to tell a teacher working at the charity what she had been going through.
She said she telephoned the teacher and opened the conversation with an explanation that she was being harassed, but the teacher ended the conversation and put the phone down before she could mention the sexual assault allegations.
Kate explained that some of those attending Kids Company with her were not "kids" but men in their 20s.
"Everyone was way older, they are all adult, not kids, they shouldn't be there. I was intimidated anyway, I felt intimidated by these men," she said.
Responding to Kate's claim, Ms Batmanghelidjh said: "I would be very concerned if a staff member turned round and said they can't do anything about it.
"I'd be very surprised and if a staff member, one staff member, had done that there were lots of others to go to."
Kids Company employs 600 paid staff, as well as working with a pool of about 8,000 volunteers and 500 students.
One woman who was a paid employee of the charity in 2009 claimed she was the victim of a sexual assault by a co-worker on a night out.
Ella (not her real name) said the man forced his hand into her underwear and had to be pulled away by people around them. She said she reported the incident to the charity, but it was never dealt with properly.
Ella said she was promised the accused co-worker would be kept away from Kids Company until Ms Batmanghelidjh had spoken to him, but the next day he was back on site.
In emails seen by the BBC, Camilla Batmanghelidjh wrote to Ella: "In relation to ******** and the police, that I would support you 100% and that in this situation you were my priority."
However, she also questioned the woman's boundaries, writing: "I'm not excusing *********'s behaviour or saying that your behaviour led to it. I'm merely asking you to be much, much clearer about establishing robust boundaries."
Ella said she felt scared that little had been done - and that the man had not been suspended and continued to work at Kids Company.
Ella herself reported the sexual assault allegation to police, but chose not to press charges.
In the Newsnight investigation, it heard that male clients of the charity's Urban Academy in their 20s had forced young clients - girls aged 16 to 18 - to have sex with them.
Referring to one such man, a former employee said: "There'd be repercussions if they didn't… you have to do it, he'd blackmail them."
She said the girls were told: "If you don't do it I'll tell them about this, this, this and this. Or [he would] photograph them naked and then threaten to send it around to other people."
This employee claims that these allegations had been raised with staff, and she knew of them herself, but said they were not passed on. Newsnight has confirmed that the allegations of coerced sex made by witnesses to the programme were not known to the authorities.
A week ago, the Metropolitan Police launched an investigation into Kids Company led by Scotland Yard's child abuse command.
Camila Batmanghelidjh has emphatically denied any wrongdoing.
She told BBC News: "We've already had lawyers go through the details.
"There isn't an allegation that will stand. I'm telling you.  Now.  In 19 years we haven't had a single Child Protection problem in the organisation.   And whenever something is raised with us we immediately report it to the police. "
Speaking to Newsnight later, Ms Batmanghelidjh said the allegations of sexual exploitation by Kids Company clients of other clients were brought to the charity's notice for the first time by police recently.
She added: "If such a thing had taken place on our premises it would have triggered all our safeguarding procedures. I have absolutely no awareness of it.
"And there is no awareness of this incident having taken place at Kids Company premises or brought to Kids Company's attention. We would have totally reported something like that."
The Kids Company charity failed in its handling of allegations of serious incidents, including sexual assaults, former staff have said.