There are an estimated 60 billion barrels of oil in the Falklands but a spokesman for Desire said the amount that would be of commericial use was probably far less.
The company told the BBC on Monday that their interest was purely in oil and sought to distance themselves from the row between London and Buenos Aires.
"Desire is an oil company and it's exploring for oil and not getting involved in what Argentina is saying about going to the UN", said David Willie, a spokesman for the company.
"The rig is sitting firmly inside [British] waters."
The Ocean Guardian, which is believed to have been taken to a point about 100km north of the islands, was recently towed from Scotland to the South Atlantic.
The company has said it will drill to an estimated target depth of about 3,500 metres in its search for oil.
The two countries' rival claims of ownership over the Falklands exploded into war in 1982 after Argentine military rulers seized the islands, only to be defeated and expelled by a British naval force.
The conflict lasted 74 days and cost the lives of 649 Argentine and 255 British military personnel.
Argentina says Britain, a permanent UN Security Council member, is skirting UN resolutions calling for dialogue on the dispute.
It says the UN resolutions recognise the territorial dispute and urge dialogue to settle it.
In January, Britain rejected Argentina's latest claim to the islands, which the UK has held and occupied since 1833.
The Falklands lie 450km off Argentina's southern coast and Buenos Aires says its territorial waters extend well beyond the archipelago, to the edge of the underwater continental shelf, more than 2,000km away.
Argentina was hoping on Monday to win more support from neighbouring countries at a regional meeting in Cancun.