The tiny archipelago, made up of two major islands and several islets of volcanic origin that house a nature reserve, is located nearly midway between Portugal's Madeira Islands and Spain's Canary Islands.
Portugal said it wants to set up an exclusive economic zone off the coast of the Savage Islands, which, under the UN Convention of the Law of the Sea, would grant the country 12 nautical miles of territorial control around them and a claim to sovereign rights to explore, exploit, conserve and manage natural resources that exist within 200 nautical miles.
But Spain argues the territory should be seen as rocks and not islands, which would give Portugal the right to just the 12 nautical miles of territorial control.
Spain has sent a letter to the UN arguing that the territory is not a small archipelago as claimed by Portugal, daily newspaper Diario de Noticias reported.
"Portugal will reiterate in writing its position before the United Nations," a Portuguese foreign ministry spokesman said without giving further details when asked about the newspaper report.
Portugal will try to reach an agreement with Spain over the territory, a Portuguese government official said.
The row over the status of the islands comes as Spain finds itself engulfed in a spat with Britain over the sovereignty of the waters around Gibraltar, a tiny British overseas territory on Spain's southern tip.
Access to the Savage Islands is restricted. Permission to visit must be granted by the Madeira Natural Park, which manages the territory.
The islands, located about 250 kilometres (150 miles) from Madeira and 165 kilometres from the Canaries, were discovered by a Portuguese navigator in the 15th century.
They were held by several wealthy Portuguese families before they were finally bought by the Portuguese state in 1971, which turned them into a nature reserve.
The Savage Islands are monitored year round by two wardens, and are visited regularly by scientists who come to study the many bird species that breed annually on its rock cliffs or stop over on their annual migrations.
Portuguese President Anibal Cavaco Silva visited the islands last year. He is the first Portuguese head of state to stay overnight on the islands as his predecessors only visited for a few hours.