Groups of young prostitutes were invited to homes and guesthouses paid for by the charity for sex parties, according to one source who claimed to have seen footage of an orgy with sex workers wearing Oxfam t-shirts.
"If they do not hand over all the information they have from their investigation subsequently to the relevant authorities including the Charity Commission and prosecuting authorities, then I can not work with them any more".
The charity said it had reported the results of its investigation to Britain's charity regulator and to major donors, including the Department for International Development.
"If the moral leadership at the top of the organisation isn't there, then we can not have you as a partner".
Oxfam said it had already taken measures to prevent sexual abuse and misconduct, including the launch of a hotline to report allegations.
Earlier Sunday Britain's International Development Secretary Penny Mordaunt warned Oxfam to cooperate with a renewed probe into the scandal or face repercussions.
The charity confirmed it had dismissed four people and allowed three others to resign in the case after an investigation uncovered offences including sexual misconduct, bullying, intimidation and failure to protect staff.
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And there was also nothing to stop them from getting former or current staff to provide a reference "in a personal capacity", she said.
"With hindsight, I would much prefer that we had talked about (the) sexual misconduct", Goldring told BBC radio.
The scandal deepened on Sunday when The Observer newspaper claimed sex workers were also used during a 2011 relief mission in Chad.
The Times of London on Friday published a report with details of a leaked internal investigation of the alleged payment for sex with young women, possibly including under-age girls, by Oxfam staff in Haiti.
The British Red Cross says it received a small number of harassment reports involving language or behaviour and that all the cases involved actions in the U.K. It said "even a small number of incidents is too many".
Oxfam will examine new complaints from staff members about how the aid workers involved in 2010 were recruited and vetted.
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Ms Thomson said she was working closely with chief executive Mark Goldring to make improvements.
Penny Mordaunt condemned the behaviour of some Oxfam staff members as a "complete betrayal", as she threatened to pull government funding ahead of the meeting.
Marr said: "That was a lie, wasn't it?"
He added: "Everybody - the 25,000 staff and volunteers - are compromised by this, the hundreds of thousands of people who support Oxfam every month are compromised by this, and to everybody I apologise".
Ms Mordaunt announced she would meet the charity on Monday to discuss the case, and said: 'If the moral leadership at the top of the organisation is not there then we cannot have you as a partner'.
She added that Oxfam had done "absolutely the wrong thing" by failing to inform authorities about the full details of the allegations.
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