The US has recently ramped up the pressure, deploying an aircraft carrier strike group and nuclear-capable bombers to counter vaguely described threats from Iran. "We have also made clear to the Iranians that if American interests are attacked we will most certainly respond in the appropriate fashion".
Huthi rebels "are launching missiles into areas where there are Russians and Americans travelling".
Pompeo - who like most of the United States political class, with the major exception of President Donald Trump, has been outspoken in his criticism of Putin - said that the two countries found areas on which to cooperate, and voiced optimism in breaking a deadlock on Syria.
Washington past year pulled out of a nuclear deal backed by Europe, Russia and China, which curbed Iran's nuclear ambitions in return for sanctions relief.
Moscow and Washington are overdue to dispel "suspicions and prejudices on both sides", Lavrov said, and to "start building a new constructive framework" of how Russian Federation and the U.S see each other.More news: Colorado School Cleared Student Violence Concerns Months Before Shooting
With tensions running high over Iran, Syria, Ukraine and Venezuela, Russian President Vladimir Putin claimed vindication from the investigation into Russia's interference in the 2016 USA presidential election and said he thought it was time to move on. Since then it has slapped sweeping sanctions on Iran.
"On Venezuela, we have disagreement", Pompeo said.
United Nations inspectors have said Iran is complying with the deal, and Moscow last week denounced new U.S. sanctions on the country's mining industry, calling for new talks to save the nuclear accord.
During their almost two-hour talk in Putin's summer home, Pompeo said they spoke at length about North Korea, weeks after Putin held a summit with leader Kim Jong Un in the Far Eastern port city of Vladivostok.
But the pair had little to agree on in their statements to the press some three hours later.More news: How much coffee is too much?
"He understands that the United States is going to be in the lead, but I think there are places we can work together", Pompeo said.
Lavrov however repeated Moscow's denials of interference and said it was "absolute fiction" that Russian Federation had colluded with the Trump campaign.
Russian Federation and the US also do not see eye to eye when it comes to Syria, where government forces have been leading a major offensive against the last rebel-held province in recent weeks.
Lavrov said the Mueller report and "baseless" allegations of Putin-Trump collusion in the 2016 election had "frozen" the relationship.
Mr Pompeo, who met with both Mr Putin, the Russian president, and Mr Lavrov during his visit, said: "Interference in American elections in unacceptable".More news: Microsoft offers security update to fix critical issue in older Windows systems
Lavrov mockingly said that the United States had also promoted democracy in war-torn Iraq and Libya and called for dialogue with Maduro - an option rejected by Washington.
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