UK Secretary of State for International Development Penny Mordaunt said the government would cut its funding for the charity unless executives could prove in a meeting with her on Monday that they had the moral leadership for it to continue.
The charity should immediately hand over its entire 2011 investigation into the sexual misconduct to the appropriate prosecuting authorities and the Charity Commission, she said.
"With regard to Oxfam and any other organization that has safeguarding issues, we expect them to cooperate fully with such authorities, and we will cease to fund any organization that does not," she said in a statement.
The warning follows reports in The Times 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, and that the UK-based charity tried to cover up the scandal at the time.
In the latest revelations, The Sunday Times reports 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 organizations."
Oxfam recorded 87 incidents last year, referring 53 to the police or authorities and dismissing 20 staff or volunteers, according to the paper.
Oxfam , which employs around 5,000 staff and has 23,000 more volunteers, received nearly 32 million pounds ($44 million) from the government in the last financial year.
Oxfam Chief Executive Mark Goldring said Saturday it receives less than 10 percent of its funding from the UK government and hoped to continue working with Downing Street while rebuilding trust with the public.
Oxfam is also accused of failing to warn other aid agencies about the staff involved, which allowed them to get jobs among vulnerable people in other disaster areas.
The UK has been struggling to deal with a series of sexual abuse scandals that have raised doubts about how institutions, including the church, sports teams and the news media respond to those who are vulnerable to abuse.
The number of abuse allegations being made in the UK has spiked since one of the BBC's top presenters, Jimmy Savile, was exposed as a serial pedophile after his death in 2011.
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