"Despite all the exotic work of Mr. Mueller's commission, I have to give him his due: on the whole, he conducted an objective investigation and confirmed the absence of any traces of a conspiracy between Russia and the current administration," Putin told Pompeo at Bocharov Ruchey, his residence outside the city. Putin said that Mueller's investigation had spoiled relations between the U.S. and Russia but that "Today I am hoping the situation is changing."
While the Mueller report found there wasn't evidence to conclude that the Trump campaign illegally conspired with Russia to meddle in the election, it did detail many interactions between members of the campaign and Russians. It also detailed in depth the extent and scope of Russia's interference in the 2016 election.
Pompeo's trip reflects a renewed outreach by the Trump administration to fulfil the president's long-held goal of establishing friendlier relations with Russia. Two weeks ago, Trump spoke with Putin by telephone in a conversation that was described as lengthy and positive by both sides. On Tuesday, Trump announced he would meet with Putin at a G20 summit in Japan next month.
Pompeo's visit -- his first to Russia as U.S. Secretary of State -- was unusually upbeat and he said he and Putin had discussed most major issues, from Venezuela, Syria, Iran and North Korea to counter-terrorism and arms control. Earlier in the day, he met with foreign minister Sergey Lavrov, holding a press conference where both sought to smooth over the harsher edges of their disagreements.
Pompeo told Lavrov Trump is "committed to improving this relationship" and that the two countries were "not destined to be adversaries on every issue."
At a joint press conference afterwards, Lavrov -- who referred to Pompeo as "Mike" throughout -- said he had concluded it was "entirely possible" to normalize relations.
Russian officials will also have been encouraged by Trump's announcement he would meet with Putin in Osako for the G20 in late June. Trump had previously said he would not meet Putin until Russia releases 24 Ukrainian sailors seized last fall. The sailors remain on trial in Russia.
Lavrov said Russia had not yet received a formal request from the White House for a meeting but that it would be pleased to do so.
But in reality, there are many issues that remain major roadblocks to a genuine rapprochement, with Russia and the U.S. on opposing sides in many crises around the world, from Venezuela to Ukraine and Syria. At home Trump also faces criticism from Democrats and Republicans who believe he has been soft on Russia's election meddling.
Asked by reporters after his meeting with Putin whether he had brought up Russian meddling in the U.S. 2016 election, Pompeo said he did.
"I said it as clearly as I could," Pompeo said.
At a press conference with Lavrov, Pompeo said he had told the Russians the U.S. would not tolerate its interference in the 2020 presidential election, warning it would put Russian-American relations in an "even worse place."
Lavrov, however, dismissed the election allegations again as "complete fiction," before launching into a lengthy discourse on historical occasions when the U.S. and Russia have accused one another of interference or pledged to avoid it, including in the 1930s.
Pompeo tried to smooth over the dispute with humor, saying, "You can see we have some disagreements on this issue. I promise not to go back to the early '30s."
Following unexplained sabotage attacks on four commercial ships in the United Arab Emirates over the weekend, mounting fears have been gathering that the crisis could slide into an inadvertent military conflict.
Those worries were intensified on Monday, after The New York Times reported the Pentagon prepared plans that would see 120,000 American troops deployed to the region if Iran or its proxies mounts an attack on American forces there.
In Sochi, Lavrov said Russia wanted to assist in ensuring the crisis with Iran did not "tip over into war." He said Pompeo had told him the report of the 120,000 troops plan was "rumors" and said he felt "the U.S. side had a commitment to finding a political solution."
Pompeo said "we fundamentally do not see a war with Iran," but warned, "We've made clear to the Iranians if American interests are attacked we most certainly will respond in an appropriate fashion."
Asked after the Russian meeting whether Iran is linked to the sabotage of the cargo ships, Pompeo said he did not yet have any concrete information.
However, he continued, “I think in the coming hours and days we'll know the answer to that but I don't have anything this evening.“
Pompeo and Lavrov were also clearly still far apart on Venezuela, where Russia has been backing president Nicolas Maduro, rejecting calls for him to step down by opposition leader Juan Guaido, who the U.S. and dozens of other countries support.
The clash between the U.S. and Russia over Venezuela has intensified in recent weeks, with Pompeo last month directly accusing Russia of blocking a deal that would have seen Maduro leave the country.
Trump, though, has publicly placed himself at odds with his advisers, including Pompeo. After their call last week, Trump said he believed "Putin is not looking at all to get involved" in Venezuela other than "he'd like to see something positive happen."
In Sochi, Pompeo said he had again told Lavrov that Russia should stop supporting Maduro. Lavrov, though, reiterated Moscow's position that there should be talks between the opposition and Maduro and compared the U.S. efforts to force him out to the previous American interventions in Iraq and Libya.
There were also diverging accounts about how much Ukraine, a major fault line between Russia and the U.S., had featured in the discussions. Pompeo said that he and Lavrov had discussed Ukraine and called on Russia to reach out to its new leader, Volodymyr Zelenskiy. But later, a top Kremlin aide, Yuri Ushakov, said Putin and Pompeo had not talked about Ukraine in their meeting.
"There was no Ukraine at all," Ushakov said.
Afterwards, Pompeo shrugged off a question about his personal take on Putin, about whom he has had sharp words in the past.
"This is about the relationship with the United States and Russia and how we move forward together, it is not about personalities," Pompeo told reporters. "Not about people it's about how do you take the interests of our two countries. We're going to protect our interests doggedly, they're going to do their best to protect their interests in that same way."
ABC News' Luis Martinez contributed to this report.