Toning down his more recent sabre-rattling rhetoric, the right-wing leader pledged to eventually return the islands to his country’s rule, but by peaceful methods.
He said it would be done through diplomatic channels and admitted there was “no instant solution”.
In a candid interview with the BBC, President Milei also conceded it could take decades to try to gain the Falklands from the UK and said Argentina would not “seek conflict”. His nation has long claimed sovereignty over the islands in the south-west Atlantic ocean – 300 miles off its coast and 8,000 miles from the UK.
But residents, who number less than 4,000, have repeatedly made it clear they wish to remain British. The UK’s Foreign Secretary, Lord Cameron, visited the islands in February and said their future was not up for discussion.
And Mr Milei said: “If that territory is now in the hands of the UK, he has a right to do that. I don’t see that as a provocation.” The Argentinian leader also praised Margaret Thatcher who was the UK prime minister during the Falklands War.
Speaking in his presidential palace office, there was even some memorabilia of the late Mrs Thatcher on a display table.
Lady Thatcher famously ordered the torpedoing of the Argentine naval cruiser the General Belgrano during the war, resulting in the deaths of 323 people on board.
But asked if he admired her, President Milei, 53, who is a free-market supporting populist, said: “Criticising someone because of their nationality or race is very intellectually precarious.
“I have heard lots of speeches by Margaret Thatcher. She was brilliant. So what’s the problem?”
The interview appeared to be a toning of his recent proclamations and came after he earlier promised a “roadmap” for the islands to become Argentine, on the 42nd anniversary of the Falklands War in early April. He had criticised politicians who “beat their chests demanding sovereignty of the islands, but without any result”.
But now Mr Milei said he wanted the islands to become Argentine “within the framework of peace”.
He added: “We are not going to relinquish our sovereignty, nor are we going to seek conflict with the UK.”
But he refused to put a timeframe on this, saying “it’s going to take time” and it would involve a “long-term negotiation”.
Asked why the UK would agree to this, he said: “They might not want to negotiate today. At some later point they might want to. Many positions have changed over time.”
He denied it was not a priority of his but, when pushed, admitted “of course” it could take decades – referencing the handover of Hong Kong from the UK to China in 1997.
The UK and Argentina went to war over the British overseas territory – known as the Malvinas in Argentina – in 1982.
The invasion claimed the lives of 255 British servicemen, three islanders and 649 Argentinian personnel.
In 2013, when Lord Cameron was prime minister, islanders voted in favour of remaining a UK overseas territory.
