The Prime Minister has insisted the matter was “settled decisively some time ago”.
During his presidential campaign, Mr Milei reportedly said Buenos Aires had "non-negotiable" sovereignty over the Falklands, the archipelago in the South Atlantic Ocean which is known as the Malvinas by Argentinians.
The islands, about 8,000 miles from the UK and roughly 300 miles from mainland Argentina, were the subject of a brief but bloody war in 1982 after Argentinian forces invaded and briefly occupied the territory.
The war claimed the lives of 255 British servicemen, three islanders and 649 Argentine personnel.
British ministers regularly cite the results of a 2013 referendum that saw close to 100 percent of voters on the islands, with its population of about 3,500 people, opt to remain a UK Overseas Territory.
Mr Milei reportedly said during a TV election debate: "What do I propose? Argentina's sovereignty over the Malvinas Islands is non-negotiable. The Malvinas are Argentine.
"Now we have to see how we are going to get them back. It is clear that the war option is not a solution. We had a war - that we lost - and now we have to make every effort to recover the islands through diplomatic channels."
In a newspaper interview with La Nacion, a daily newspaper in Argentina, Mr Milei proposed that the UK hand over the Falklands to his South American country in a similar way to how Hong Kong was given back over to Chinese rule in 1997.
The populist politician, who has described himself as an anarcho-capitalist and is said to have lauded Margaret Thatcher - the British prime minister during the Falklands conflict - said the views of those living on the islands "cannot be ignored".
Speaking earlier today, the Prime Minister's official spokesman said: "The UK has no doubt about the sovereignty of the Falkland Islands, and indeed South Georgia and the South Sandwich Islands.
"The UK Government will continue to proactively defend the Falkland islanders' right to self determination."
The spokesman said Falklands rule was an "issue that was settled decisively some time ago".
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Since the war, more than 40 years ago, UK ministers have been keen to shoot down any talk of Argentina having any claim to the islands.
In July, Mr Sunak hit out at the European Union's "regrettable choice of words" after it appeared to have endorsed the name Argentina prefers.
A diplomatic row erupted after Brussels referred to the islands as "Islas Malvinas" in a declaration jointly signed with Argentina and other Latin American countries.
His latest remarks were also backed by Defence Secretary Grant Shapps, who earlier today tweeted: "The Falkland Islands are British. That is non-negotiable and undeniable.
"99.8 percent of islanders voted to remain British and we will always defend their right to self-determination and the UK's sovereignty."