Ties between the two countries have been poisoned by allegations - denied by Moscow - that Russian Federation tried to influence the results of the 2016 USA presidential election, and by differences over Venezuela, Iran, Syria, and Ukraine.
"If Russia engaged in that in 2020, it would put our relationship in an even worse place than it has been".
North Korean leader Kim Jong Un last month met Russia's Vladimir Putin for their first face-to-face talks.
In a key point of contention, Pompeo has rebuffed demands from North Korea, which has the backing of Russian Federation and China, for an early lifting of sanctions before a final deal.
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.More news: Manchester City ‘fully co-operating in good faith’ with UEFA investigation
Despite Trump's conciliatory attitude toward Putin, Washington and Moscow are at loggerheads on an array of urgent strategic questions, including the Venezuelan crisis, Iran, the Syrian civil war and the conflict in Ukraine.
In the highest-level face-to-face contact between the two countries since special counsel Robert Mueller's report was released last month, Putin told Pompeo he hoped relations with the USA would now improve.
It's a odd comment, to say the least, considering that Mueller did establish that Russian Federation meddled in the 2016 election to benefit Trump.
European counterparts urged Pompeo Monday to de-escalate tensions in the Gulf after Washington sent an aircraft carrier and bombers to protect its interests and President Donald Trump said Tehran would "suffer greatly". Instead, he said, Iran has "chosen the path of resistance", adding that USA officials know it's not in their interest to engage in a hot military conflict.
Pompeo told reporters after his talks with Putin that the two sides had a "very productive" conversation on Syria and how to move the political process there forward and were able to "make some truly constructive process points" on Afghanistan. Russian Federation has interests in oil projects in Venezuela and has propped up Mr Maduro's regime with loans.More news: Bob Iger Teases A Potential Third Live-Action Star Wars Series
Tuesday's Sochi meeting came after Pompeo shared intelligence and details with European allies about what the United States calls Iran's recent "escalating threat", blaming Tehran for failing to choose talks over threats.
"Despite the exotic nature of Mr Mueller's commission, on the whole he conducted quite an objective investigation and confirmed the absence of any collusion between the U.S. administration and Russia", Putin said.
Responding to Lavrov's opening remarks, Pompeo said: "I'm here today because President Trump is committed to improving this relationship".
Russian Federation and the USA 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.More news: HTC releases smartphone with Bitcoin full node capabilities