Jorge Taiana will meet Mr Ban at 3.30pm New York time (8.30pm British time), the office of the spokesman for the Secretary-General said.
The Argentinian foreign minister requested the meeting.
Meanwhile Argentina's president spoke out at a regional summit, reiterating the country's sovereignty claim to the islands and blaming the Foreign Office for adding to tensions.
Today's Times newspaper said a resolution was also set to be tabled in the UN General Assembly condemning Britain for allowing Desire Petroleum's Ocean Guardian to begin drilling 60 miles north of the islands after Argentina announced new shipping controls.
Defence Minister Bill Rammell said the Government will take "whatever steps are necessary" to protect the islands, which had a "legitimate right" to develop an oil industry within their waters.
Ocean Guardian was towed 8,000 miles from Scotland to the south Atlantic isles for the exploration.
Argentina formally objected to the move and tightened shipping regulations in the region.
Latin American and Caribbean nations backed Argentina's claim of sovereignty to the islands as the dispute gathered pace.
At the Rio Group summit of 32 countries hosted by Mexico, Argentina presented a statement quoting Mexican President Felipe Calderon as saying that "the heads of state represented here reaffirm their support for the legitimate rights of the republic of Argentina in the sovereignty dispute with Great Britain".
Argentinian president Cristina Fernandez ruled out any plans to try to keep the British boats or rigs out. "We do not believe in methods like blockades," she told reporters.
However in a statement to regional leaders in Cancun, Mexico she reiterated Argentina's claim to the islands and blamed the Foreign Office for fanning tensions.
An English translation of her speech was posted on the official website of Argentina's presidency.
In the statement, Ms Fernandez criticised the Foreign Office.
She said: "When in these last few days England decided to install an oil rig offshore and Argentina - in exercise of its domestic law - made decisions as a Sovereign State, the Foreign Office floated the idea of a potential war threat by Argentina. I would say that is ridiculous, cynical because few countries have shown more signs of peaceful intentions than Argentina after the advent of democracy.
"Our armed forces only participate in joint UN peace missions in Haiti and Cyprus. We are not present in Afghanistan or Iraq; we are opposed to any sort of occupation; to any sort of violation of international law because we believe this is one of the reasons for a world that is becoming more insecure, dangerous, fragmented and not because of ideological disputes - which as already mentioned characterised the 20th century - but because of more serious matters such as religions or deeper interests as the handling by each of our countries of the right to own and use our renewable and nonrenewable natural resources."
She said diplomatic efforts to establish sovereignty would continue, saying: "Argentina will insist on its democratic vocation of fully respecting international law in its claim and shall adopt, within the framework of its domestic law, all provisions and resolutions that tend to reassert our sovereignty over the Southern Archipelagos. This is no whim of this President or of previous presidents but a requirement of our National Constitution.
"In 1994 when the National Constitution was amended I had the honour of acting as a Constitutional Reform Assembly member. On that occasion, our Constitution was made to enshrine the obligation of all governments to continue claiming and reaffirming our sovereignty over the Malvinas Islands."
Desire said the well is being drilled to an estimated target depth of about 3,500 metres (11,500 feet), and drilling operations are expected to take about 30 days.
The company said the rig is sitting "firmly inside UK waters".
Argentina claims the Falklands - known as Las Malvinas in Buenos Aires - are occupied by Britain illegally.
Argentina invaded the Falklands in 1982, before a UK taskforce made up of some 100 ships seized them back in a war which claimed the lives of 255 British servicemen and women and more than 600 Argentinians.
Brazilian President Luiz Inacio Lula da Silva criticised the United Nations for not pushing more forcefully to reopen the debate on the Falkland Islands.
"What is the geographic, the political or economic explanation for the UK to be in Las Malvinas?" Mr da Silva asked.
"Could it be because the UK is a permanent member of the UN's Security Council where they can do everything and the others nothing?"
Jan Cheek, a member of the Falklands legislative assembly, said she was "not surprised" by the Latin American countries' statement, as they had historically tended to back Argentina in the dispute.
Ms Cheek told BBC Radio 4's Today programme: "We don't feel isolated, because we don't rely too heavily on Latin America for many of our supplies or much of our communication.
"I would much prefer it if we could treat all our Latin American neighbours as good friendly neighbours, but under the present Argentine regime I don't think that will happen.
"Argentina does regularly take this issue to the UN but it has very little effect on us, mainly because what they are seeking is to make us a colony of Argentina while we have freely chosen, through self-determination, to be an overseas territory of the UK."
Ms Cheek said there is "no reason" why Argentina's stance should affect the search for oil, adding: "It certainly makes no difference to our right to license the natural resources of our territory".
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