French President Francois Hollande has said allegations that the US bugged European embassies could threaten a huge planned EU-US trade deal.
He said there could be no negotiations without guarantees that spying would stop "immediately".
US Secretary of State John Kerry said he did not know the truth of the claims but sought to down play them.
Meanwhile, Russian and US security agencies are reportedly discussing how to deal with the man behind the leaks.
Former CIA-analyst Edward Snowden is believed to be at an airport in Moscow, seeking a destination safe from the US where he is wanted for prosecution over the leaking of thousands of classified documents.
Russia's President Vladimir Putin and US's President Barack Obama have ordered the chiefs of their respective agencies, FSB and FBI, to find a way out of the impasse, a senior Russian official said.
The allegations that US security services bugged EU missions and the embassies of friendly European countries - including the French, Italian and Greek embassies - were published at the weekend by Der Spiegel in Germany and the Guardian in Britain.
The claims have angered many in Europe.
The European Commission called it "disturbing news if proven true" and said it expected "clarity and transparency" about the issue from Washington.
German Chancellor Angela Merkel's spokesman, Steffen Seibert, said "bugging friends is unacceptable... we are no longer in the Cold War".
Italy's Foreign Minister Emma Bonino said Rome had requested from Washington "clarification of a very thorny affair".
Talks over the EU-US pact, the biggest bilateral deal ever negotiated, are due to start in Washington DC on 8 July.
France only cleared the way for the talks in mid-June, after EU members accepted its demand to shield movies and online entertainment from the might of Hollywood and Silicon Valley.
But France's President Hollande signalled on Monday that the negotiations could be further delayed if the US cannot give a guarantee it had ended its surveillance of the EU.
"We cannot accept this kind of behaviour between partners and allies. We ask that this immediately stop," he told journalists during a visit to western France.
"There can be no negotiations or transactions in all areas until we have obtained these guarantees, for France but also for all of the European Union, for all partners of the United States."
Steffen Seibert has said that Germany wants the deal to go ahead but "mutual trust is necessary in order to come to an agreement".
John Kerry said he did not know the truth of the allegations, but that he had been asked about them by the EU's foreign policy chief Catherine Ashton and would report back to her.
But at a news conference in Brunei, he said: "Every country in the world that is engaged in international affairs of national security undertakes lots of activities to protect its national security and all kinds of information contributes to that.
"And all I know is that is not unusual for lots of nations. But beyond that I'm not going to comment any further until I have all the facts and find out precisely what the situation is."
Edward Snowden has been charged in the US with theft of government property, unauthorised communication of national defence information and wilful communication of classified communications intelligence.
He left Hong Kong after revealing his identity, and is reportedly staying at an airport hotel in Moscow from where he has applied for asylum in Ecuador.
Green parties in France and Germany on Monday called on their governments to offer Mr Snowden asylum.
"Someone like that should be protected," said Juergen Trittin, leader of Germany's Greens.
"He should get safe haven here in Europe because he has done us a service by revealing a massive attack on European citizens and companies. Germany, as part of Europe, could do that."
Green Party leaders have also called for existing US-EU agreements on the exchange of bank transfer and passenger record information to be cancelled.