The scandal-hit charity will be confronted at a showdown meeting on Monday, following the revelations about misconduct and possibly abuse in Haiti following the 2010 earthquake.
“If the moral leadership at the top of the organisation is not there, we cannot have you as a partner,” Ms Mordaunt said.
She added: “If they do not hand over all the information that they have, from their investigation and subsequently, to the relevant authorities – including the Charity Commission and the prosecuting authorities – then I cannot work with them any more as an aid agency.”
Oxfam would be given a last “opportunity to tell me their side of the story, so I have all the facts”, she told the BBC’s The Andrew Marr Show.
The charity, which received nearly £32m from the Government in the last financial year, has denied a cover-up over the events in Haiti and promised full co-operation with authorities.
Its own investigation into the allegations led to four people being sacked and three others resigning.
However, Oxfam is now facing further allegations, reported in The Observer, that its staff also used prostitutes in Chad in 2006 – a mission led by the same senior staff member, Roland van Hauwermeiren, who resigned from Oxfam in 2011.
During the interview, Ms Mordaunt suggested Oxfam had lied and had failed in its “moral leadership” – condemning the behaviour of some staff members as a “complete betrayal”.
The charity had “categorically” stated to the Department for International Development (DfID) that no harm had been done in Haiti and that beneficiaries were not involved.
When Mr Marr said: “That was a lie, wasn’t it?“, Ms Mordaunt replied: “Well, quite.”
She added that Oxfam had done “absolutely the wrong thing” by failing to inform authorities about the full details of the allegations.
The Charity Commission said that it had written to Oxfam “as a matter of urgency” to request further information.
The regulator said an Oxfam report on the investigation stated there had been no allegations of abuse of beneficiaries and made no mention of any potential sexual crimes involving minors.
“Our approach to this matter would have been different had the full details that have been reported been disclosed to us at the time,” the Commission said in a statement.
Oxfam has said that allegations that under-age girls may have been involved were not proven.
The UK-based charity said it had already set up a whistleblowing hotline to prevent sexual abuse and misconduct.
The International Development Secretary has written to all charities, warning they will have funding withdrawn if they fail to comply with authorities over safeguarding issues.
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