It was mostly smiles and conviviality in Sochi on Tuesday as the US secretary of state, Mike Pompeo, and Russian foreign minister, Sergei Lavrov, hinted at a warming in relations ahead of further talks including Vladimir Putin.

“These were good, intensive, substantive, useful talks,” Mr Lavrov said at an evening press conference. “I hope the visit will not only help improve the atmosphere but also help resolution … on bilateral, regional and international issues.”

Mr Pompeo, who earlier described his trip as “a good faith effort”, returned the compliments.

The conversation was a “good step”, he said, towards his president’s “goal of better relations”. There had been “progress” on delivering a political resolution in Syria; “agreement” over Korean and denuclearisation; a “resolution” to work on a “broader, verifiable and enforceable” framework on arms control. The United States also left “an open door” on the possibility of extending current strategic arms agreements.

But, the secretary of state was keen to point out, serious disagreements remained.

Trump in Helsinki: Putin meeting and protests Show all 16 1 /16 Trump in Helsinki: Putin meeting and protests Trump in Helsinki: Putin meeting and protests President Trump meets with Russian President Vladimir Putin in Helsinki Reuters Trump in Helsinki: Putin meeting and protests An advert from Finnish newspaper Helsingin Sanomat is on display in Helsinki Reuters Trump in Helsinki: Putin meeting and protests President Trump in a meeting with Finnish President Sauli Niinisto AFP/Getty Trump in Helsinki: Putin meeting and protests President Trump meets with Finnish President Sauli Niinisto in Helsinki EPA Trump in Helsinki: Putin meeting and protests President Trump talks with Finnish President Sauli Niinisto in Mantyniemi, the official residence of the Finnish President EPA Trump in Helsinki: Putin meeting and protests US, Finnish and Russian flags fly in front of the Presidential Palace in Helsinki EPA Trump in Helsinki: Putin meeting and protests A protester wears a mask featuring a portrait of Russian President Vladimir Putin during the so-called "Helsinki against Trump and Putin" demonstration on July 16, 2018, in the Finnish capital Helsinki. The US and Russian leaders opened an historic summit in Helsinki on Monday, with Donald Trump promising an "extraordinary relationship" and Vladimir Putin saying it was high time to thrash out disputes around the world. / AFP PHOTO / Jonathan NACKSTRANDJONATHAN NACKSTRAND/AFP/Getty Images JONATHAN NACKSTRAND AFP/Getty Trump in Helsinki: Putin meeting and protests Vladimir Putin gifts a football to President Trump at the press conference that followed their meeting AP Trump in Helsinki: Putin meeting and protests Trump supporters hold banners during a demonstration in Helsinki AFP/Getty Trump in Helsinki: Putin meeting and protests People protest for women's reproductive rights in Helsinki's Senate Square Reuters Trump in Helsinki: Putin meeting and protests A crowd watches the motorcade transporting President Trump through Helsinki AFP/Getty Trump in Helsinki: Putin meeting and protests Russian President Putin drives through Helsinki on his way to meet with President Trump EPA Trump in Helsinki: Putin meeting and protests A man in the audience of the joint press conference holds up a sign sign that reads "NUCLEAR WEAPON BAN TREATY". REUTERS Trump in Helsinki: Putin meeting and protests A man is removed from the joint press conference in Helsinki. Security removed the man after he pulled out a sign that read "NUCLEAR WEAPON BAN TREATY". REUTERS Trump in Helsinki: Putin meeting and protests President Trump and Russia's President Putin sit for a working lunch in Finland's Presidential Palace AFP/Getty Trump in Helsinki: Putin meeting and protests President Trump meets with Russian President Vladimir Putin in Helsinki Reuters

On Venezuela, and Russian support for Nicholas Maduro, the two countries saw things “very differently”. Mr Maduro had brought “nothing but misery to the Venezuelan people”, he said. The sides would not agree on Ukraine either – and “could never support the Russian attempts to annex Crimea".

On both issues, however, the US secretary of state was careful to dangle a carrot.

In Venezuela, Russia had business interests that would have to be “considered” under a new democratic regime, he said, hinting at a deal to protect Russian state oil company Rosneft’s billions of dollars of oil investments.

In Ukraine, sanctions “remained in place.” But they could be “regulated” were Russia were to “reach out to Ukraine’s new president”, Volodymyr Zelensky. The prospect of Russia releasing the Ukrainian servicemen captured near Kerch in December 2018 was meanwhile mentioned as something that might help regulation.

As expected, one of the major discussion topics was the prospect of military confrontation between the US and Iran following the apparent breakdown of the 2015 Iranian nuclear deal. That simmering crisis had caused Mr Pompeo to cancel a scheduled Moscow leg of the tour on Monday in order to visit Brussels and address concerned European colleagues.

Earlier in the day, a Kremlin spokesman underlined the Russian position on the matter: that Washington’s decision to pull out of the framework was “reckless".

“Putin has talked about his lack of comprehension for such a policy,” said Dmitry Peskov. “It leads to hounding a country into a corner. As recent history tells us, this never gets a good result.”

Mr Pompeo brought up the elephant in the room: claims of Russian interference in US domestic politics. It was time for Russia to show “those kinds of activities are a thing of the past”, he said.

Mr Lavrov seemed happy at the invitation to wax historical. Allegations of Russian interference were made “throughout the ages,” he said. They had even appeared in the 1986 presidential campaign, "where the presidential ambitions of a certain Donald Trump were first reported." Maybe the United States would agree to finally sign a mutual agreement not to interfere in domestic politics first proposed by the Russians in the 1930s?

“I promised I would not speak about the 1930s,” Mr Pompeo answered.

On arrival in Sochi, Mr Pompeo said that he had been tasked with improving bilateral relations and finding areas of agreement.

While not quite of the level of the Stalinist era, US-Russia ties are undoubtedly the worst they have been for several decades, battered by wars, distrust, sanctions and counter-sanctions. Links do exist, but they are limited. Besides counter-terrorism, Russian-US dialogue is only effective at the level of the heads of Security Councils and in policy discussion of certain regional issues such as Syria, North Korea and Afghanistan.

Mr Lavrov said the talks offered a prospect of a normalisation of relations across government.

“It’s time to construct a new, more constructive matrix of understanding each other," he said. "We are ready.”

The grounds for a thaw came earlier this month following a 3 May telephone call between Presidents Trump and Putin. Kremlin sources described the call as “unexpectedly friendly”. Donald Trump meanwhile took to Twitter to suggest the same: ”Tremendous potential for a good/great relationship with Russia.”

Since then Mr Pompeo and Mr Lavrov met at the fringes of the Arctic Council in Helsinki. And, on Monday, Mr Trump unexpectedly announced he was ready to meet Mr Putin at June’s G20 meeting in Osaka in June.

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Initially, there was some confusion and contradictory signals as to how the Russians were disposed to the proposals. But at press conference Mr Lavrov seemed to confirm that they would accept — but only if it the offer was made formally.

“We have received information that Mr Trump is ready to meet,” he said. “If we get an official proposal… we will respond positively.”

The emphasis on protocol was not accidental, says Andrey Sushentsov, programme director of the Valdai Club, a discussion forum with close links to the government.

Moscow has become weary of contradictory messages from Washington, he suggested.

“The foreign policy process is totally disorganised. For Venezuela, John Bolton is taking the lead. For Syria, it’s the Pentagon. Pompeo is in charge of North Korea. Trump is obsessed about Iran and China. And Congress is suspicious about any foreign initiative of the administration,” he said.