The US Secretary of State said America would not get involved in the dispute itself but was ready to mediate.
Mrs Clinton said the United States was willing to be a go-between if it would help resolve differences which have resurface due to British plans to drill for oil in the region.
“It is our position that this is a matter to be resolved between the United Kingdom and Argentina,” she said. “If we can be of any help in facilitating such an effort, we stand ready to do that.”
President Cristina Kirchner of Argentina is expected to ask Mrs Clinton to put pressure on Britain to discuss the Falklands Islands when the pair meet in Buenos Aires on Monday night.
Mrs Clinton made a brief trip to the Argentine capital to meet the Argentine president, a personal friend of the Clintons, at the beginning of a week-long tour of Latin America.
Barack Obama has been criticised in Britain for assuming a neutral position in the renewed tensions with Argentina after a UK oil rig began drilling last week in disputed waters off the north coast of the South Atlantic islands.
Spain, currently holding the European presidency, said the European Union should not become involved in the Falklands dispute.
While Spain has historically supported the Argentine claim, Madrid believes that the dispute should remain a “bilateral matter and not one between regions”, said officials.
Mrs Clinton’s brief meeting with Mrs Kirchner was reportedly scaled back after the Argentine president told CNN that Mr Obama had been a disappointment in South America.
She has also criticised his administration’s response to the recent coup in Honduras and caused annoyance when she spent the day of Mr Obama’s inauguration with Fidel Castro in Cuba.
Mrs Clinton’s tour will also take in Chile, currently in chaos following an 8.8 magnitude earthquake, as well as Uruguay, Brazil, Costa Rica and Guatemala.
As with previous Argentine politicians, President Kirchner has been accused of re-stoking the Falklands row to distract attention from domestic problems.
She persuaded a summit meeting last week of 32 Latin American and Caribbean countries to issue a brief statement supporting Argentina’s sovereignty claim.
Argentina is also once more pushing in the United Nations for Britain to be called to negotiate over the islands’ future.