The Argentinian government has issued a decree that tightens control over shipping in the area ahead of British efforts to start oil and gas exploration off the islands' waters.
According to The Sun newspaper, a new British naval task force comprising two ships and an oil supply tanker has been sent to the South Atlantic isles.
But the Ministry of Defence (MoD) said it was "maintaining" British force levels in and around the Falkland Islands and Mr Brown said he did not expect to need to send a task force to the area.
Speaking on a phone-in with Real Radio in the North East, he said: "I think you will find we have made all the preparations that are necessary to make sure the Falkland Islanders are properly protected.
"...This is oil drilling that is exploration for the future. It is perfectly within our rights to do this."
He added that he believed the Argentinian government understood this and "sensible discussions" would prevail.
Shadow foreign secretary William Hague suggested that an increased naval presence in the area might be necessary.
He said: "The islanders have every right to develop the natural resources of their islands and surrounding waters.
"It should be made very clear that the wholly legitimate search for oil in the Falklands waters will not be affected by unwarranted threats or interference from Argentina.
"Additionally, the British Government should state clearly that no vessel operating within the territorial waters of the Falkland Islands will require any form of permit from any other country.
"Increased British naval presence in the area would leave no doubt as to this position."
The MoD said the UK's existing "deterrence" in the Falklands was not being increased and UK force levels were being maintained.
A spokesman said: "The Government is fully committed to the South Atlantic Overseas Territories, which include the Falkland Islands.
"A deterrence force is maintained on the islands. That deterrence force comprises a wide range of land, air and maritime assets which collectively maintain our defence posture.
"We have a permanent presence in the South Atlantic, including one frigate/destroyer, a patrol vessel, a survey ship and a replenishment vessel.
"We also have 1,076 service personnel on land."
Argentina's president Cristina Fernandez de Kirchner issued a decree covering all ships that enter its waters, which Buenos Aires regards as those covering the entire South Atlantic continental shelf.
The Foreign Office shrugged off the demand yesterday, insisting that the seas around the Falklands were controlled by island authorities and would not be affected by the decree.
But it is understood to be watching the situation closely.
As the issue escalated, Sir Nicholas Winterton, chairman of the all-party Falklands group, said he would seek a meeting with senior officials at the Foreign Office when Parliament returns from recess next week.
He dismissed Ms Fernandez's decree as "pathetic and useless" as Argentina has no jurisdiction over the seas around the Falklands.
And he stressed that both the Government and Conservative opposition remain committed to British sovereignty over the islands and the principle of self-determination for their inhabitants.
"The Argentinians are again indulging in hostile behaviour - albeit at this stage only in words - against a friendly neighbour, the Falklands," said Sir Nicholas.
"I believe they are doing so for internal purposes and that it will not affect the Falkland Islands at all.
"All they are trying to do is impede the economic progress of the Falkland Islands because, of course, the encouragement of hydrocarbon exploration in the area is an important part of achieving a sustainable future for the islands.
"I don't think one wants to exacerbate what is already a difficult situation, but clearly it is important that the Foreign Office indicates that they believe that this decree has no jurisdiction over international waters."
The Foreign Office said Britain was ready to co-operate with Argentina on South Atlantic issues and was working to develop relations between the two countries.
"Argentina and the UK are important partners," said the Foreign Office spokesman.
"We have a close and productive relationship on a range of bilateral and multilateral issues, including the global economic situation (particularly in the G20), human rights, climate change, sustainable development and counter-proliferation.
"And we want, and have offered, to co-operate on South Atlantic issues. We will work to develop this relationship further."
Despite being ejected from the islands, which they know as the Malvinas, after a two-month occupation and war in 1982, Argentina continues to claim sovereignty over the Falklands.
The dispute over the territory has flared up recently because of interest in possible oil reserves.
Several British companies are poised to begin exploration using an offshore rig, while Desire Petroleum has licensed six areas where it predicts 3.5 billion barrels of oil and nine trillion cubic feet of natural gas can be recovered.
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