In a further warning to the charity, she told the BBC: "If they do not hand over all the information they have from their investigation and subsequently to the relevant authorities, including the Charity Commission and prosecuting authorities, then I can not work with them any more as an aid delivery partner".
According to The Observer, women were repeatedly invited "for parties" at Oxfam's team premises.
Caroline Thomson, the chair of Oxfam Great Britain's board of trustees, says charities that work in "fragile and unstable environments can become targets for abusers".
Ms Mordaunt said the charity had also "categorically" stated to the Department for International Development (DfID) that no harm was done and beneficiaries were not involved.
If the moral leadership at the top of the organisation is not there then we can not have you as a partner.
A spokesperson for Oxfam said the organisation was deeply shocked by the new allegations, admitting the problem was "sector-wide".
Worldwide development secretary Penny Mordaunt said it was "despicable" that sexual exploitation and abuse existed in the aid sector and threatened to cut funding to charities if they fail to comply with authorities over safeguarding issues.
She challenged Oxfam bosses to explain why a resulting investigation saw four staff dismissed and three resign - with no public disclosure.
And there was also nothing to stop them from getting former or current staff to provide a reference "in a personal capacity", she said. "We were not told about the nature of these events".
He is expected to insist that Oxfam did not inform ministers of the.
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Responding to a report in The Times newspaper, Oxfam admitted that the behaviour of some of its staff had been "totally unacceptable".
It said one of the instances was "a case of failings in adequate safeguarding", which was reported to the Charity Commission.
Some Oxfam workers, including its most senior mission director, have been accused of paying survivors for sex while there.
Oxfam, one of the world's most prominent relief agencies, could lose its funding from the British government over reports that its workers exploited survivors of a massive quake in Haiti, and possibly other disasters, for sex.
"Unfortunately, there is nothing we can do to stop individuals falsifying references, getting others that were dismissed to act as referees and claiming it was a reference from Oxfam", a spokeswoman added.
There had been other incidents since 2011, he said, but "nothing on the scale of this severity".
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".
Marr said: 'That was a lie, wasn't it?'
"It doesn't matter whether you've got a whistleblowing hotline, it doesn't matter if you've got good safeguarding practices in place".