Earlier on Wednesday, the US State Department ordered the departure of non-emergency US government employees from its embassy in Baghdad and consulate-general in Erbil, as well as the suspension of normal visa services.
US Iranian envoy Brian Hook told reporters travelling with Mr Pompeo in Brussels that the secretary of state shared intelligence on Iran with allies since "Europe shares our concerns about stability in the Gulf and the Middle East".
Capitol Hill is paying close attention.
Germany and the Netherlands have halted all training missions in Iraq as tensions continue to rise between the United States and Iran, which supports proxy militias there.
The State Department warns that "the US government's ability to provide routine and emergency services to USA citizens in Iraq is extremely limited".
Foreign Minister Mohammad Javad Zarif was speaking in a meeting with his Japanese counterpart as tensions rise in the Middle East, fuelling concern that the United States and Iran are heading for conflict.
Also on Wednesday, Germany - which has 160 soldiers in Iraq - suspended military training operations, citing increasing regional tensions, while the Netherlands suspended a mission providing assistance to Iraqi local authorities, Dutch news agency ANP said.
That same day, a top British general in the US-led operation to combat the Islamic State (IS) group in Iraq and Syria blew a hole in the Trump administration's assertion that Iran-allied militias were mobilising to attack USA forces. Since last week, House leaders have been asking for a classified briefing for lawmakers on the situation with Iran, but Pelosi said the administration indicated it couldn't come together "that fast".
Though U.S. allies have contradicted American warnings about the Iranian threat, Trump and his senior advisers have maintained their bellicose stance.
Recent days have seen allegations of sabotage attacks targeting oil tankers off the coast of the United Arab Emirates, a drone attack on a Saudi oil pipeline claimed by Yemen's Iranian-allied Houthi rebels, and the dispatch of US warships and bombers to the region.
Iran's top security body dismissed this as "psychological warfare".
"We fundamentally do not seek war with Iran", Pompeo said during a visit to Sochi, Russia, this week.
Former CIA chief John Brennan responded by tweeting, "Your refusal to accept the unanimous assessment of U.S. Intelligence on Iran, No. Korea, ISIS, Russia, & so much more shows the extent of your intellectual bankruptcy. If we did that, we'd send a hell of a lot more troops than that".
It was also reported on Tuesday that U.S. investigators believed Iran or groups it supported had used explosives to damage four tankers off the coast of the United Arab Emirates on Sunday.
Tensions between the two countries have been souring ever since Trump took office three years ago, especially since the US President withdrew from the nuclear deal with the Islamic Republic.
The US in recent days has ordered the USS Abraham Lincoln aircraft carrier strike group to the Gulf region, plus four B-52 bombers.
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