Kerry went straight from four days of intensive diplomacy in the Middle East to a meeting hosted by Southeast Asian nations in the gas-rich sultanate of Brunei, where South China Sea disputes and North Korea were also set to dominate talks.
Kerry was due to hold a series of direct meetings with counterparts from world powers on the sidelines of the forum, which ends tomorrow with a gathering of 26 Asia-Pacific countries and the European Union.
European allies have demanded answers from the United States on the claims, reported by German weekly Der Spiegel, and warned relations would be damaged if the allegations proved to be true.
"We can't negotiate a large transatlantic market if there is any doubt that our partners are bugging the offices of European negotiators," EU Justice Commissioner Viviane Reding said.
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One document, dated September 2010 and classed as "strictly confidential", describes how the National Security Agency kept tabs on the EU's mission in Washington, Der Spiegel said.
The United States said yesterday it would respond to the EU via diplomatic channels over the bugging allegations.
The Snowden affair is also likely to come up when Kerry holds talks with Russian Foreign Minister Sergei Lavrov in Brunei tomorrow.
Washington is angry that Snowden, a former government contractor wanted by the United States after divulging details of the widespread surveillance on communications, flew to Moscow from Hong Kong as he sought asylum, possibly in Ecuador.
Russia has refused to hand Snowden over.
"I'm actually anxious to get there (Brunei) and to engage with him because the situation in Syria is grave," Kerry told reporters in Tel Aviv before flying to Asia.