European parliament president Martin Schulz led the condemnation, following claims over the weekend by US Prism whistle blower Edward Snowden in the German weekly news magazine Der Spiegel that the NSA had routinely bugged EU officials and offices in Brussels, New York and Washington.
"I am deeply worried and shocked about the allegations of US authorities spying on EU offices", said Schulz on Saturday, adding, "If the allegations prove to be true, it would be an extremely serious matter which will have a severe impact on EU-US relations.
"On behalf of the European parliament, I demand full clarification and require further information speedily from the US authorities with regard to these allegations."
EU foreign policy chief Catherine Ashton's reaction was more cautious, saying that she was "aware" of the allegations that EU premises have been under US surveillance.
"As soon as we saw these [press] reports, the European external action service made contact with the US authorities in both Washington DC and Brussels to seek urgent clarification of the veracity of the facts surrounding these allegations.
"The US authorities have told us they are checking on the accuracy of the information released [on Sunday] and will come back to us as soon as possible.
"While this is clearly a matter for concern, we will make no further comments at this stage, until we have more clarity on the matter."
However, the new allegations - coming just a few weeks after it was revealed that the NSA was also involved in secret surveillance that gathered EU citizen user data from internet giants such as Google and Microsoft - left MEPs in a much more retaliatory mood.
ALDE group leader Guy Verhofstadt said, "If confirmed, these revelations are absolutely unacceptable and must be stopped immediately", adding that, "The Americans must come clean with their EU partners about their covert snooping. It undermines trust in a strategic partnership at a crucial moment in transatlantic relations."
The revelations could stall the beginning of negotiations on a new free trade deal with Washington.
Announced last month and heralded as a "game changer" for both the EU and US by commission president José Manuel Barroso, Verhofstadt said, "I cannot see how a crucial trade and investment partnership can be concluded as long as this spectre of spying hangs over us."
The former Belgian prime minister added, "Barely a week goes by now without some new shocking evidence of intrusive data mining by the US national security agency."
"It poses some very uncomfortable questions for us all. We need urgent reassurances from our US partners. Parliament must launch its own inquiry into the matter."
The co-presidents of parliament's Greens/EFA group, Rebecca Harms and Dany Cohn-Bendit added their weight to an official inquiry, saying they would use this week's special plenary debate on the NSA's activities to call for an investigation.
"A special committee of the EU parliament must be launched immediately", said Harms, adding, "We must not postpone the decision to after the summer break."
"Before starting negotiations on a future trade agreement with the US, we need a debate about the infringement of international law by Prism and Tempora. The last few days have shown how urgently we need an international agreement on data protection."
Cohn-Bendit, meanwhile, called for the EU to grant asylum to Edward Snowden, a former contractor for the CIA, because of the fact that he had, "stood up for human and civil rights, especially those of European citizens".
"Granting him asylum now is a question of European values. To leave him stuck in limbo in a no-man's land is ignoble and damages the EU's image."
Cohn-Bendit also revealed that the Greens/EFA group would propose Snowden for the next European parliament's Sakharov prize. The annual prize for freedom of thought honours individuals or organisations for their efforts in protecting human rights and fundamental freedoms.
Further reaction came from the chair of parliament's influential economic and monetary affairs committee, Sharon Bowles, who also warned that the revelations could have a serious impact on the planned trade talks.
"The upcoming EU-US trade talks will certainly be under strain in the wake of this scandal. There will be an enormous elephant in the room which will be impossible to ignore," said Bowles.
"I will be leading a delegation of my committee to the US in a couple of weeks and the issue of bugging is bound to come up. The last time I was in Washington I was briefed in a coffee shop, although this was due to a very early start. That said, it might now be safer to conduct all our business in coffee shops."
Dutch deputy Sophie In't Veld, a vice chair of parliament's civil liberties committee said, "The spying scandal puts Europe to the test. Now Europe has to show its citizens it is able and willing to protect the rights of European citizens.
"We need to ask the US for clarification. It is unlikely we will get the full facts, but the US government will have to explain their actions to the EU authorities. Not in closed sessions of an expert group, but at the highest political level.
"We have repeatedly urged the European commission to ensure that EU laws are not overruled by foreign laws on our own EU territory. We can no longer accept the extraterritorial application of US law or the law of any other country on EU territory."
The issue is likely to dominate parliament's planned debate this Wednesday in Strasbourg on the activities of the NSA and its impact on EU data privacy.