Ecologists want to highlight the threat in the run-up to a meeting in Croatia of the International Commission for the Conservation of Atlantic Tunas.
And they have made common cause with small-scale fishermen who see industrial-scale practices ravaging the stocks on which they base their livelihood.
The international commission is to reevaluate annual fishing quotes for the first time since 2002, which are currently fixed at 32,000 tons for the "western Atlantic" zone.
Prior to the November meeting International Commission for the Conservation of Atlantic Tunas scientists will meet in Madrid from June 12 to 18 to draw up a list of recommendations.
Greenpeace and the WWF hope to make their own voices heard.
"In 2002, scientists recommended 22,000 tons, but (the International Commission for the Conservation of Atlantic Tunas) decided to give 10,000 tons more," said Garcia.
Even this generous quota has been exceeded, say environmentalists, who claim the overall catch in 2005 hit between 40,000 and 50,000 tons due to fishermen over-raiding stocks.
"This time we want to see the quotas come down enormously in response to the crisis," said Garcia.
The environmentalists are not targeting the small-scale fishermen who use nets according to local custom, but industrial flotillas and their high-tech operations.
With boats guided by satellite and planes that circle above banks of tuna during the breeding season, they capture thousands at one go before taking them to tuna farms to be fattened.
The centers, which receive large-scale European Union subsidies, are farms in name only as they do not practice any kind of aquaculture, red tuna being unable to reproduce in captivity.