The Cabinet minister announced she will meet with the British-based charity's leadership on Monday as it continues to reel from accusations it covered up the episode in 2011.
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"If they do not hand over all the information that they have from their investigation... then I cannot work with them any more as an aid delivery partner -- and any other organisation in those circumstances," she told the BBC.
In a sign of widening fallout from the scandal, the Sunday Times reported more than 120 workers for Britain's leading charities were accused of sexual abuse in the past year, "fuelling fears paedophiles are targeting overseas aid organisations".
Mordaunt said she was writing to all charities that receive state aid demanding they step up efforts to tackle sexual misconduct among staff or face funding cut-offs.
She will also seek to confirm they have referred all concerns about specific cases and individuals to the relevant authorities.
"I think this is an emerging picture, there are more allegations coming out about Oxfam and other organisations... and all of them will be followed up," Mordaunt vowed.
'No cover-up'
The government's hardening stance follows reports in The Times newspaper that young sex workers were hired by Oxfam's senior staff in Haiti after the 2010 earthquake which devastated the island and left up to 300,000 people dead.
The charity -- which employs around 5,000 staff and has 23,000 more volunteers -- recorded 87 incidents last year, referring 53 to the police or authorities and dismissing 20 staff or volunteers, according to the Sunday Times.
Oxfam Chief Executive Mark Goldring admitted Saturday that it had failed to detail fully the nature of the 2011 scandal but insisted it "did anything but cover it up".
"With hindsight, I would much prefer that we had talked about (the) sexual misconduct," Goldring told BBC radio.
"But I don't think it was in anyone's best interest to be describing the details of the behaviour in a way that was actually going to draw extreme attention to it."
However, Mordaunt said not disclosing the full picture was "a scandal" and Monday's meeting was a chance "to see if they are displaying the moral leadership that I think they need to now".
She added: "what is so disturbing about Oxfam is that when this was reported to them, they completely failed to do the right thing."
The minister noted offenses committed by British citizens anywhere in the world could still be prosecuted in Britain.
"We're talking about a historic case, but it is in some respects still live.
"They still have information they should be giving to the authorities."
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