A special focus, according to Der Spiegel, was Germany, Europe's biggest economy. Quoting NSA documents, the magazine said that every month, half a billion phone calls, emails, text messages and internet chat entries were siphoned off Germany's phone system and internet.
Analysts say the row has been fuelled by the notion of massive spying by an ally and disappointment with Obama. In some respects it is an unprecedented public bust-up, exceeding the outrage sparked by George W. Bush's America First policies. For now, though, it will not affect the strategic relationship, they say. And they note a disingenuousness in the declarations of shock and outrage.
"Everyone spies on everyone else, but Americans spy on more people," quipped a senior figure in a European intelligence agency.
Last month, Britain was accused of setting set up a fake internet cafe at two G20 summits in 2009 to access phone calls and emails from allied politicians.
The everyone-does-it argument was made indirectly by Obama, but he also promised Washington would respond to allies' questions. "In European capitals there are people who are interested, if not in what I had for breakfast, at least what my talking points would be if I am talking to their leaders."
Suspicions that the US has eavesdropped for political or economic gain, rather than for fighting terrorism or organised crime, will erode its "soft power" - the store of goodwill accumulated through World War II and the Cold War.
Revulsion in Germany is strong, given its Orwellian experiences with the Gestapo and the Stasi. "It's beyond our imagination that our friends in the US consider the Europeans as enemies," said Justice Minister Sabine Leutheusser-Schnarrenberger. "If the media reports are accurate, it is reminiscent of actions among enemies during the Cold War."
If Hollande's threat is sincere, there is a de-facto deadline for next Tuesday, when talks gets under way in Washington for a Transatlantic Trade and Investment Partnership, or TTIP.
German Chancellor Angela Merkel's spokesman said Berlin was keen on the talks but stressed "mutual trust is necessary to come to an agreement".
Trade in goods between the US and the 27-country EU last year was worth €500 billion ($834 billion), with another €280 billion in services and trillions in investment flows. The TTIP would be the world's biggest bilateral trade pact, adding €119 billion annually to the EU economy, and €95 billion for the US.
The sticking point for the TTIP could be the European Parliament, which under EU law must approve any treaty between the EU and other parties, and is less vulnerable to US pressures than national governments.
Even before the scandal, the assembly had taken the lead in opposing unfettered access to European data, such as details of airline passengers, by US agencies.
It also blocked an agreement on sharing information about bank transfers to track suspected terrorists until the deal incorporated further safeguards.
Transatlantic Trade and Investment Partnership
Trade
€500b trade between the US and the EU
€280b in services between the two
Pact
€119bwould be added annually to the EU
€95bwould be added annually for the US.