Susana Malcorra, Argentina's foreign minister, is said to be supported for the prestige role by both US President Barack Obama and Susan Rice, his national security adviser.
The prospect of an Argentine head of the UN comes against the continuing dispute over the Falkland islands, which both Argentina and the UN say must be "decolonised" and returned to Argentina.
Washington has refused to back the Falklanders' right to self-determination, despite a 2013 referendum where 99.8 per cent of the islanders voted in favour of remaining a British Overseas Territory.
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The role of UN Secretary-General will be vacated by Ban Ki-moon on December 31.
Malcorra, aged 61, only declared her candidacy last month but has been enthusiastically backed by the Americans.
The UK is said to be keen for a woman to head the UN for the first time in its 70 years, but some MEPs have expressed concerns about an Argentine candidacy.
ECR group member Charles Tannock told this website, "It would be deeply problematic if the UN Secretary General promoted a domestic policy of her country of origin which violated the sovereignty of a UNSC permanent member the UK and the principle of self-determination."
Tannock, a member of Parliament's foreign affairs committee, added, "She would have in my personal view to give absolute guarantees never to raise this matter under her tenure for her to be acceptable to the UK and the Falklanders."
Further concern was voiced by UKIP leader Nigel Farage, who said, "I said a few weeks ago that Barack Obama was the most anti-British President in the history of the US. I am surprised that this assertion was re-confirmed in such a short time. The USA acts in its own interests."
Speaking recently, Malcorra said, "Argentina believes there is an opportunity to improve the relation with the Falkland Islanders, and we are exploring available options, but those links must be channelled through bilateral relations with the UK since the islands dispute is with the UK."
She added the Argentine ministry was working with some ideas, "and as soon as those ideas have been developed we are going to share them, but we don't want to anticipate anything since it's a complex, delicate issue, and we want to gear it as it should."
Tensions over the Falklands still crackle more than 30 years after Argentine forces seized the islands in 1982 and Britain sent a task force to retake them in a brief war which saw more than 600 Argentine and 255 British servicemen killed.
The Argentine government claims the islands, which are 300 miles off the Argentine coast and 8000 miles from Britain, as its own and has stepped up a campaign to get what it calls Las Malvinas back as exploration by oil and gas firms nearby has raised diplomatic tensions.