US relations with Russia “may be at an all-time low”, Donald Trump has warned, as he moved even further away from his campaign promise to establish better ties with Moscow.

“Right now we’re not getting along with Russia at all,” the US President said, following talks on Syria in the Russian capital between the US Secretary of State and his counterpart Sergey Lavrov.

Rex Tillerson earlier spoke in similar terms to Mr Trump with a warning that trust between Moscow and Washington has plunged to a “low level” amid rising tensions over the war in Syria.

America’s top diplomat said the world’s “two foremost nuclear powers cannot have this kind of relationship” during a tense joint news conference with the Russian foreign minister.

Mr Lavrov warned of attempts by unnamed groups to sabotage ties with the US and said he and Mr Tillerson had covered a whole host of issues including Syria, Ukraine, and allegations of Russia’s interference in the 2016 US presidential election.

Speaking in Washington, Mr Trump said he thought the discussions had been “very successful”.

Mr Tillerson also met with Russian President Vladimir Putin for nearly two hours on Wednesday.

Ahead of the off-camera meeting, Mr Putin noted in an interview that “one could say that the level of trust on a working level, especially on the military level, has not improved, but rather has deteriorated”.

Tensions flared recently when the US, after warning Russian forces in the area, launched nearly 60 missiles into Syria in response to a suspected sarin gas attack by President Bashar al-Assad on civilians in Idlib province.

Mr Tillerson said the US was “quite confident” that the Assad regime was responsible.

Both he and Mr Lavrov agreed during their meeting in Moscow that an investigation into the incident should take place and that it should be conducted by the Organisation for the Prohibition of Chemical Weapons, an international body monitoring the use of such weapons in countries that signed a 1997 treaty.

In pictures: US missile strike against Syria Show all 7 1 /7 In pictures: US missile strike against Syria In pictures: US missile strike against Syria The guided-missile destroyer USS Porter (DDG 78) launches a tomahawk land attack missile in the Mediterranean Sea AP In pictures: US missile strike against Syria The United States military launched at least 50 tomahawk cruise missiles at al-Shayrat military airfield near Homs, Syria, in response to the Syrian military's alleged use of chemical weapons in an airstrike in a rebel held area in Idlib province EPA In pictures: US missile strike against Syria Shayrat airfield in Syria Getty Images In pictures: US missile strike against Syria US Navy guided-missile destroyer USS Ross (DDG 71) fires a tomahawk land attack missile in Mediterranean Sea Reuters In pictures: US missile strike against Syria US Navy guided-missile destroyer USS Ross (DDG 71) fires a tomahawk land attack missile in Mediterranean Sea Reuters In pictures: US missile strike against Syria President Donald Trump speaks at Mar-a-Lago in Palm Beach, Fla., after the US fired a barrage of cruise missiles into Syria in retaliation for this week's gruesome chemical weapons attack against civilians AP In pictures: US missile strike against Syria Syria's President Bashar al-Assad Reuters

Mr Lavrov said Russia was urging Mr Assad, as the “legitimate government”, to give experts access to the chemical weapons attack site as well as the airfield where it is said to have originated.

The Russian foreign minister noted during the news conference that his country was not disputing that chemical weapons were used, but language should not be used that assumes the Assad regime is responsible without a proper investigation.

He repeatedly stressed that the investigation should be “impartial” and “objective”.

Mr Tillerson said that although the Assad regime should leave power in an “orderly way”, it was up to the “political process” determined by the Syrian people to determine exactly how that happens.

Russia as “an ally has the best means of making [Mr Assad] understand this reality”, Mr Tillerson said.

Tillerson listens to a journalist’s question during a news conference with the Russian foreign minister (AP) (AP Photo/Ivan Sekretarev)

Last week’s attack in Idlib province resulted in nearly 100 deaths, with horrific video footage emerging of the victims – many of them children.

Though there were several points of disagreement in the talks between Mr Tillerson and his Russian counterpart, both diplomats said there were areas of common ground and further discussions should take place.

Mr Lavrov said the Kremlin was willing to reinstate an air safety agreement aimed at preventing accidents between US and Russian jets. The agreement was put on hold in the wake of the US missile strike.

Mr Tillerson also noted that the two sides agreed to set up a “working group” in order to “stabilise” the relationship between the two nations and be able to discuss larger issues.

He did not say what specific topics this special group, consisting of foreign ministry staffers from both countries, would discuss.

However, several thorny topics remain on the table, including Ukraine, which Mr Tillerson noted was essential to strengthening US-Russia ties.

Then there is the allegations of Russian cyber criminals interfering in the US presidential election through the hacking of the Democratic National Committee offices as well as the rampant spread of fake news sites that may have negatively affected the public perception of Mr Trump’s opponent Hillary Clinton.

It is widely noted that Mr Putin and Ms Clinton had an icy relationship when she was Secretary of State.

Mr Lavrov said repeatedly that the US has not shown any evidence of Russian tampering to the Kremlin on the matter and that several questions remain unanswered.

He once again blamed the Obama administration for their lack of action on a special project to combat cyber crime that Russia had approached the US with 18 months ago.

“We still have the project on the table,” Mr Lavrov said.

He did agree though that the overall discussion was “productive”, as Mr Tillerson put it.

“The common threat is obvious,” said Mr Lavrov, referring to combating terror attacks on both Americans and Russians.

Last week’s suspected chemical attack also prompted a strong response from US Ambassador to the United Nations, Nikki Haley, in front of the Security Council, who said the Russians were either “incompetent” or were “played for fools” by the Assad regime.

Vladimir Safronov, a Russian representative to the UN, said there was no evidence to prove the Assad regime were the ones to carry out the attack since some stockpiles were controlled by opposition groups.

However, US Secretary of Defence James Mattis said there was “no doubt” it was the Assad regime based on US intelligence.