President Trump has asked his national security adviser, John Bolton, to extend the invitation for a “working level” dialogue between the two leaders.

Mr Trump was roundly criticised by Democrats and Republicans in Washington for siding with the Russian leader over US intelligence reports that Moscow interfered in the 2016 presidential election. The president has had to reverse or clarify his comments a number of times since, amplifying the fallout from his joint press conference alongside Mr Putin. There have also been concerns in Congress about the private nature of the one-on-one meetings between Mr Trump and Mr Putin

Even Mr Trump’s director of national intelligence, Dan Coats, said he did not know what happened in Helsinki. “Well, you’re right, I don’t know what happened at that meeting,” Mr Coats said in response to a question at the Aspen Security Forum in Colorado.

The coveted invitation to Washington was sure to be seen as a victory by Mr Putin, whose last official visit to the United States was in July 2007, when he spent two days at the Bush family compound in Kennebunkport, Maine.

The US president had tweeted his meeting with Mr Putin had been a “great success”, despite the largely negative reaction, and that he was looking forward to a second meeting “so that we can start implementing some of the many things discussed, including stopping terrorism, security for Israel, nuclear proliferation, cyber attacks, trade, Ukraine, Middle East peace, North Korea and more.” He gave no indication of when that second meeting might take place.

But on Thursday afternoon, White House press secretary Sarah Huckabee Sanders revealed negotiations were underway to invite the Russian leader to Washington later this year.

“President Trump asked @Ambjohnbolton to invite President Putin to Washington in the fall and those discussions are already underway,” she tweeted.

Trump claims that Russia is no longer targeting the US

Top Senate Democrat Chuck Schumer immediately criticised the planned invitation. “Until we know what happened at that two hour meeting in Helsinki, the president should have no more one-on-one interactions with Putin. In the United States, in Russia, or anywhere else,” he said in a statement.

The last official visit by a Russian president to the United States was in June 2010, when Dmitri Medvedev, now Russian prime minister, visited the United States.

The offer comes after the White House said Mr Trump had rejected Mr Putin’s proposal that Russian authorities be allowed to question American citizens, after the offer drew fierce criticism in the United States, one of at least three clarifications.

During a news conference following their meeting, Mr Putin described the proposal when he was asked about the possible extradition of 12 Russian intelligence officers indicted three days previously on charges of meddling in the 2016 presidential election.

Mr Putin indicated he would permit American law enforcement officials to observe questioning by Russian officials of the indicted Russians in exchange for letting Russian investigators question Americans on other matters, mentioning London-based financier Bill Browder, a one-time investor in Russia. Mr Trump had called Mr Putin’s idea “an incredible offer” during the same news conference.

“It is a proposal that was made in sincerity by President Putin, but President Trump disagrees with it,” Ms Sanders said in a statement earlier on Thursday. “Hopefully President Putin will have the 12 identified Russians come to the United States to prove their innocence or guilt.”

On Tuesday, Mr Trump said he “misspoke” during the news conference and said that he did accept US intelligence service reports that Russia meddled in the election. On Wednesday, the president answered “no” to a reporter’s question on whether Russia was still targeting the United States, only to have Ms Sanders say later he was saying “no” to answering any questions – not to the question itself.

On Wednesday, the Russian Prosecutor General’s office listed Americans it wanted to question for “illegal activities,” including Michael McFaul, who was US ambassador to Russia under Democratic former President Barack Obama. “That’s not going to happen,” US Secretary of State Mike Pompeo told CBN News.

In a rebuke to the president, the Senate – controlled by Mr Trump’s fellow Republicans – unanimously approved a resolution on Thursday expressing congressional opposition to any US officials being questioned with Russian officials.

There was no immediate response from Russia to Mr Trump’s invitation. Earlier, in Moscow, Mr Putin had accused forces in the US of seeking to undermine the success of the two leaders first bilateral summit. He vowed that he and Mr Trump would regardless work to improve relations between the world’s most heavily-armed nuclear powers.

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

Speaking to Russian diplomats from around the world who had gathered Mr Putin said his meeting on Monday with Mr Trump had been “successful overall and led to some useful agreements”.

“Of course, let’s see how events will develop further,” he said, according to Reuters.

He claimed powerful forces were trying to derail any improvement in relations between the two countries.

“We see that there are forces in the United States that are prepared to casually sacrifice Russian-US relations, to sacrifice them for their ambitions in an internal political battle in the United States,” he said.

Mr Putin did not single out any individuals by name but spoke of US politicians who put their “narrow party interests” above the best interests of their country.

In truth, while Mr Trump’s performance in Helsinki infuriated Republicans and Democrats alike when he appeared to question the judgement of the US intelligence community and failed to press Mr Putin on the 2016 interference, most criticism was very muted.

While critics such as senators John McCain and Jeff Flake – one of whom has cancer and while the other is not seeking re-election – were strident in their criticism, most members of Mr Trump’s party either kept quiet or spoke in very guarded terms. Many are aware they are likely to need the support of Mr Trump and his supporters if they want to win in November’s midterms elections.

Mr Putin’s speech, while high on threats, contained few new departures. Criticism of Nato enlargement has been a cornerstone of his foreign policy rhetoric, especially in later terms.

“He wasn’t signalling change,” Mark Galeotti, Senior Non-Resident Fellow at the Institute of International Relations in Prague, told The Independent. “What this was, perhaps, was a more explicit trolling of those in the US who are opposed to better relations with Russia.”

He added: “It shows that while the Russians are happy with Helsinki, they don’t really think anything has changed.”

In his speech, Mr Putin warned of the dangers of Moscow and Washington failing to continue to mend ties, saying the New START strategic arms reduction treaty would expire soon unless both countries took action.

“Despite the difference in opinions [with Trump], we did agree that Russian-US relations are in an extremely unsatisfactory state,” said the Russian leader. “In many respects, they are even worse than during the Cold War.”

There has been some confusion about what precisely the two leaders discussed in a two-hour meeting, when no one else was present except for the US president’s interpreter.

After the pair’s private meeting, Russian Ministry of Defence spokesman Igor Konashenkov told the Tass news agency the department was “ready for practical implementation of the agreements in the sphere of international security reached by” the two leaders.

He went on to say the ministry was also “ready to enliven contact with the US colleagues, between our General Staffs and via other communication channels, to discuss extension of a nuclear arms treaty, cooperation in Syria, and other topical issues of military security”.

In the US, however, top US military leaders, including Defence Secretary James Mattis, have not addressed the matter.