France and Germany are pushing for talks with the US to find a new “understanding” by the year’s end.
A number of allegations against US intelligence agents have surfaced this week, including the bugging of German Chancellor Angela Merkel’s phone.
In addition there have been claims that the US National Security Agency (NSA) monitored millions of French telephone calls.
On Thursday, the UK’s Guardian newspaper also reported that it had obtained a confidential memo from the NSA suggesting it had monitored the phones of 35 world leaders.
The latest revelations have been sourced to US whistleblower Edward Snowden, the former intelligence contractor who fled the country earlier this year and is now in Russia.
They have overshadowed other issues at the EU summit in Brussels, including the Mediterranean migration problem, which frames the agenda of Friday’s talks.
Italian authorities said they had intercepted some 800 migrants trying to cross the Mediterranean as the EU leaders prepared to meet.
The statement of heads of state or government, released on Friday, reflects the EU leaders’ conclusions following their talks on Thursday.
It says the recent intelligence issues had raised “deep concerns” among European citizens.
The statement says the leaders “underlined the close relationship between Europe and the USA and the value of that partnership”.
It continues: “[The leaders] stressed that intelligence-gathering is a vital element in the fight against terrorism.”
And it went on: “A lack of trust could prejudice the necessary cooperation in the field of intelligence-gathering.”
Talks among the EU leaders had continued late into Thursday night.
Speaking afterwards, Mrs Merkel said: “We need trust among allies and partners. Such trust now has to be built anew.
“The United States of America and Europe face common challenges. We are allies. But such an alliance can only be built on trust. That’s why I repeat again: Spying among friends – that cannot be.”
Germany and France said they were proposing talks with the US to settle the row by the end of the year. But Mrs Merkel stressed Berlin and Paris would be making their own separate approaches to Washington.
The leaders’ statement said: “The heads of state or government took note of the intention of France and Germany to seek bilateral talks with the USA with the aim of finding before the end of the year an understanding on mutual relations in that field.
“They noted that other EU countries are welcome to join this initiative.”
French President Francois Hollande said on Friday; “What is at stake is preserving our relations with the United States.”
BBC Europe editor Gavin Hewitt says the French and Germans are looking for a new set of rules with a “no spying pact” at the core.
He says this would mirror an arrangement the United States has had with Britain, Australia, New Zealand and Canada since just after World War II. That secret intelligence-sharing operation is known as Five Eyes.
UK Prime Minister David Cameron acknowledged the UK’s “unique partnership” with the US regarding security but said: “I understand what others want to do and support that, as I think does President Obama”.
Angela Merkel said she did not know the exact details of the Five Eyes arrangement so could not say whether that is exactly what Germany is seeking, but she said: “We need something clear-cut, in line with the spirit of an alliance”.
A number of other leaders have also indicated their support for the French and German position.
Belgian Prime Minister Elio di Rupo said: “The objective must remain the same – to fight against terrorism but also respect privacy.
“Everyone can understand the need for exceptional measures given the danger of terrorism… but we are not in the position where we should spy on each other.”
Finnish PM Jyrki Katainen said: “We have to talk together with the Americans, and try to find some sort of code of conduct [on] how to cooperate on this kind of issue in the future.”
Swedish PM Fredrik Reinfeldt said it was “completely unacceptable” to eavesdrop on the leader of an ally, a view echoed by Italian PM Enrico Letta, who added: “We want the truth.”
But the BBC’s Chris Morris in Brussels says that, despite the widespread anger about American spying, Mrs Merkel opposed a suggestion to suspend trade talks with the United States – and on that point, the UK will be relieved.
Other leaders signalled the need to move on.
Lithuanian President Dalia Grybauskaite said: “The main thing is that we look to the future. The trans-Atlantic partnership was, and is, important.”
Mrs Merkel had raised her concerns with US President Barack Obama in a call on Wednesday.
White House spokesman Jay Carney later said Mr Obama had assured Mrs Merkel that her phone was not being listened to now and would not be in the future.
However, his statement left open the question of whether calls had been listened to in the past.