Spain has urged the US to give details of any eavesdropping, amid reports it monitored 60 million Spanish telephone calls in a month.
The US ambassador to Spain, who had been summoned by its EU minister, vowed to clear the "doubts" that had arisen about his country's alleged espionage.
The minister, Inigo Mendez de Vigo, said such practices, if true, were "inappropriate and unacceptable".
An EU delegate in Washington said there had been "a breakdown of trust".
Representatives from the European Parliament's Committee on Civil Liberties, Justice and Home Affairs spoke to members of the US Congress about the alleged US spying on European leaders and citizens.
"We wanted to transmit to them first that this mass surveillance of EU citizens is a genuine concern," British Labour MEP Claude Moraes, a member of the delegation, told the BBC after the meetings.
"I think they're listening to that. They want to have some sort of dialogue."
Spanish press review
Editorial in El Mundo: "The massive spying on Spanish citizens requires a strong response from the authorities... The foreign ministry should raise a formal complaint. Mariano Rajoy should join France and Germany in their initiatives. And as early as [Monday], the public prosecutor should denounce the NSA 'for violation of the privacy of millions of Spaniards, which is punishable by up to four years in prison under Article 197 of the Penal Code'."
Editorial in La Vanguardia: "The erosion of transatlantic confidence is evident and there may be significant consequences for the future of democracy unless a solution is found more or less immediately... In the coming days, a European delegation will visit President Obama to ask him about the background to the espionage. That should be the time to rebuild lost trust in each other."
Editorial in ABC: "Obama is not the first president to spy on leaders of other countries, including allies. What makes Obama different is that he is facing a terrorist threat of colossal dimensions, he is taking advantage of new technologies and he is a victim of the fragility of a world in which information flows unchecked and within seconds all secrets are revealed."
But Mr Moraes said he and his fellow delegates were unsatisfied with the "stock" responses from US officials on the issue.
"They're giving us answers, but not the answers we want," he said. "We're getting a bit tired of this, 'Well, spying has always existed.'"
German Chancellor Angela Merkel is also sending intelligence officials to Washington to demand answers to claims that her phones were tapped for a decade.
German media reported that the US had bugged Ms Merkel's phone for more than a decade - and that the surveillance only ended a few months ago.
The German government hoped that trust between the two countries could be restored, a spokesman told a news conference in Berlin.
"It would be disturbing if these suspicions turned out to be true. But Germany and the United States can solve this problem together," Steffen Seibert said.
"We will vigorously push ahead with the clarification of this case especially because we have a great interest in a good German-American relationships."
Meanwhile, a Japanese news agency says the US National Security Agency (NSA) asked the Japanese government in 2011 to help it monitor fibre-optic cables carrying personal data through Japan, to the Asia-Pacific region.
The reports, carried by Kyodo, say that this was intended to allow the US to spy on China, but that Japanese officials refused, citing legal restrictions and a shortage of personnel.Contents 'not recorded'
The latest allegation, published by Spain's El Mundo newspaper, is that the NSA tracked tens of millions of phone calls, texts and emails of Spanish citizens, in December 2012 and January 2013. The monitoring allegedly peaked on 11 December.
The White House has so far declined to comment on the El Mundo report.
It is not clear how the alleged surveillance was carried out, whether it was through monitoring fibre-optic cables, data obtained from telecommunication companies, or other means.
The NSA is reported to have collected the sender and recipient addresses of emails, along with their IP addresses, the message file size, and sometimes the top or subject line of the message.
For each telephone call, the numbers of the caller and recipient are believed to have been logged, as was its duration, time, date and location.
The contents of the telephone call itself, however, were not monitored, US intelligence officials say. The NSA has also suggested it does not usually store the geolocational information of mobile phone calls, which could determined by noting which mobile signal towers were used.