But Britain has ruled out any concessions involving the territory which it has held since the early 1800s.
The issue exploded anew this week when Cristina Kirchner, Argentina's president, protested against the start of British oil drilling near the Falklands, calling it "robbery".
Chris Bryant, the British Foreign Office minister, responded to her comments by saying: "We have no doubt about our sovereignty over the Falkland Islands and we're clear that the Falkland Islands government is entitled to develop a hydrocarbons industry within its waters."
He added that Argentinian protests were unnecessary as London was ready to work with Buenos Aires on the oil issue.
"We mustn't lose sight of the fact that Argentina and the UK are important partners. We have a close and productive relationship on a range of issues: in the G20, on climate change, sustainable development and counter-proliferation," he was quoted as saying by the Telegraph newspaper in Britain.
Argentina's call for negotiations won support this week at a summit of Latin American and Caribbean countries in Mexico.
The 32 leaders said they backed "the legitimate rights" of Argentina, although they stopped short of directly endorsing its claim to sovereignty.
But Jan Cheek, a member of the islands' legislative assembly, has accused Argentina of colonialism.
"It seems to many of us that Argentina is indulging in a little latter day colonialism in ignoring our right to self-determination and seeking to make us a colony of Argentina," he told the Associated Press news agency.