The right-wing leader toned down his more recent saber-rattling rhetoric and ultimately promised to return the islands to his country's rule, but peacefully.
He said this would be done through diplomatic channels and admitted there was “no immediate solution”.
In a candid interview with the BBCPresident Milei also admitted that it could take decades to try to win the Falklands from Britain and said Argentina would “not seek conflict”. His country has long claimed sovereignty over the islands in the southwestern Atlantic Ocean – 300 miles off the coast and 8,000 miles from Britain.
But its residents, who number fewer than 4,000, have repeatedly made it clear they want to remain British. British Foreign Secretary Lord Cameron visited the islands in February and said their future was not in question.
And Mr Milei said: “If that area is now in the hands of Britain, he has the right to do that. I don't see that as a provocation.” The Argentine leader also praised Margaret Thatcher, the British Prime Minister during the Falklands War.
When he spoke in the office of his presidential palace, there was even some of the late Mrs Thatcher's memorabilia on a display table.
During the war, Lady Thatcher famously ordered the torpedoing of the Argentine naval cruiser General Belgrano, killing 323 people on board.
But when asked if he admired her, 53-year-old President Milei, a populist who supports the free market, said: “Criticizing someone because of their nationality or race is intellectually very precarious.
“I have heard many speeches by Margaret Thatcher. She was brilliant. What's the problem?”
The interview appeared to complement his recent proclamations and came after he previously promised a “road map” for the islands to become Argentinian, on the 42nd anniversary of the Falklands War in early April. He had criticized politicians who “beat their chests and demanded sovereignty over the islands, but without any results”.
But now Milei said he wanted the islands to become Argentinian “within the framework of peace.”
He added: “We are not going to give up our sovereignty, nor are we going to seek conflict with Britain.”
But he refused to put a timetable on this, saying “it will take time” and “long-term negotiations” would be needed.
Asked why Britain would agree to this, he said: “They may not want to negotiate today. They might want to do that at a later date. Many positions have changed over time.”
He denied that this was not a priority of his, but when pressed, “of course” conceded that it could take decades – referring to the 1997 handover of Hong Kong from Britain to China.
Britain and Argentina went to war in 1982 over the British Overseas Territory – known in Argentina as the Malvinas.
The invasion claimed the lives of 255 British soldiers, three islanders and 649 Argentine personnel.
In 2013, when Lord Cameron was Prime Minister, the islanders voted to remain a British Overseas Territory.