The EU is demanding assurances that Europeans' rights are not being infringed by massive, newly revealed US surveillance programmes.
Last week a series of leaks by a former CIA worker led to claims the US had a vast surveillance network with much less oversight than previously thought.
The US insists its snooping is legal under domestic law.
The Obama administration is investigating whether the disclosures by former CIA worker Edward Snowden were a criminal offence.
More revelations promised
Mr Snowden's employer, defence contractor Booz Allen Hamilton, said on Tuesday it had fired the 29-year-old infrastructure analyst for violating its ethics code.
US officials say the snooping programme known as Prism, revealed in last week's leaks, is authorised under the Foreign Intelligence Surveillance Act (Fisa).
But details about the individuals targeted under the act remain secret, and there are concerns the NSA is overstepping its powers.
Documents leaked to the Guardian and Washington Post newspapers claimed the US authorities had direct access to the servers of nine major US technology firms, including Apple, Facebook and Google.
Mr Snowden told the Guardian that individual operatives had the power to tap into anyone's emails at any time.
Although the firms have denied granting such access, saying they agreed only to legal requests, US officials have admitted Prism exists.
And on Tuesday, Republican Senator Lindsay Graham said US surveillance of phone records allowed the government to monitor phone records for a pattern of calls, even if those numbers had no known connection to terrorism.
Russia 'consider' asylumOne of the Guardian journalists who wrote the Prism stories, Glenn Greenwald, has promised "more significant revelations" to come.
In the US, the controversy has focused on the possibility that conversations of US citizens may inadvertently be captured.
But overseas, governments and activists point out that US law provides foreigners with no protection.
Justice Commissioner Reding tweeted: "This case shows why a clear legal framework for the protection of personal data is not a luxury but a necessity."
In the US, Mitch McConnell, the top Republican in the Senate, said the person responsible for the leak should be prosecuted to the fullest extent of the law.
He said privacy concerns were understandable, given the scope of the programmes, but added it was hard to comprehend why Mr Snowden would give information to US enemies.
Senator Dianne Feinstein, the chairwoman of the Senate intelligence committee, said the American authorities were "aggressively" pursuing him.
The California Democrat also accused Mr Snowden of "an act of treason".
The top Republican in the US House of Representatives, Speaker John Boehner, labelled Mr Snowden a "traitor".
In other developments on Tuesday:
- A spokesman for Russian President Vladimir Putin said Russia would consider granting asylum to the American, should he request it
- Google asked the justice department to release every government information request to prove it did not give officials "unfettered access" to user data
- The American Civil Liberties Union (ACLU), a liberal advocacy group, has filed a lawsuit against the Obama administration challenging the legality of its phone surveillance programme
- Director of National Intelligence James Clapper has sent a message to tens of thousands of US intelligence workers reiterating the need to keep classified documents secret, Reuters reports.
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