Theresa May and Donald Trump have agreed a "window of opportunity now exists" to persuade Russia that its links with Bashar Assad's regime are no longer in its strategic interest as they discussed the situation in Syria.
The two leaders spoke about the crisis by telephone as the US president thanked the PM for her backing of his military strikes against Syrian government forces last week.
A Downing Street spokesman said: "Theresa May spoke with US President Trump to discuss last week's chemical weapons attack in Syria and the US response.
"The president thanked the Prime Minister for her support in the wake of last week's US military action against the Assad regime.
"The Prime Minister and the president agreed that a window of opportunity now exists in which to persuade Russia that its alliance with Assad is no longer in its strategic interest.
"They agreed that US Secretary of State (Rex) Tillerson's visit to Moscow this week provides an opportunity to make progress towards a solution which will deliver a lasting political settlement.
"They also discussed the broader Middle East, including the threat posed by Iran throughout the region.
"The Prime Minister and president also stressed the importance of the international community, including China, putting pressure on North Korea to constrain the threat it poses."
The Prime Minister, who is on a short walking holiday in Wales, is being kept up to date on events in Syria.
Mr Trump ordered a series of missile strikes last week in response to the deaths of more than 80 people, including children, during a chemical attack in the rebel-held town of Khan Sheikhoun.
The US president's spokesman Sean Spicer warned on Monday that further action would be considered in certain circumstances.
"When you watch babies and children being gassed, and suffer under barrel bombs, you are instantaneously moved to action. I think this president has made it very clear that if those actions were to continue, further action will definitely be considered by the United States."
The move came as Foreign Secretary Boris Johnson warned that senior Russian military officers involved in co-ordinating President Assad's campaign of repression against his own people could face international sanctions.
At a meeting of G7 foreign ministers in Lucca, Italy, Mr Johnson issued a fresh appeal to Russian President Vladimir Putin to abandon his backing for his Syrian ally following last week's chemical weapons attack.
He said if the Russians continued to support the regime in Damascus, they would be "contaminated" by its actions and could find themselves the target of new international sanctions.
"We will be discussing the possibility of further sanctions certainly on some of the Syrian military figures and indeed on some of the Russian military figures who have been involved in co-ordinating the Syrian military efforts and are thereby contaminated by the appalling behaviour of the Assad regime," Mr Johnson told reporters.
The Foreign Secretary said Mr Trump's decision to launch cruise missiles against a Syrian air base in response to the regime's use of sarin nerve agent had "changed the game" and the Russians had to decide which side they were on.
"They have a choice. That choice is to stick like glue to the Assad regime - that toxic regime which poisons its own people and is indeed poisoning the reputation of Russia - or to work with the rest of world to find a political solution," he said.
Mr Johnson defended his decision to pull out of talks with his Russian counterpart Sergey Lavrov, saying it was important that Mr Tillerson had the "clearest possible mandate" when he went to Moscow to deliver the response of the G7.
"It think it is very important in these circumstances for the world to present a united front and for there to be absolutely no ambiguity about the message," he said.
Earlier, Downing Street reiterated the Government's support for the US missile strike while calling for a renewed diplomatic push to end Syria's six-year civil war.
Mrs May's official spokesman refused to be drawn on whether the UK would support further US military action, saying it was a "hypothetical question".
Russia and Iran - Assad's two principal international backers - warned on Sunday that they would respond "with force" to any fresh attack on their ally.
President Trump's son Eric said that the US would not be "pushed around" by Mr Putin, the Daily Telegraph reports.
The president is not intimidated by Moscow's talk of war, his son said.
He insisted there was "no one harder" than his father if Russia chose to "cross us".
"If they disrespect us, and if they cross us, fine. There will be no one harder - he has got more backbone than anybody.
"He is not a guy who gets intimidated. I can tell you he is tough and he won't be pushed around. The cards will shake out the way they do, but he's tough."
The president was influenced in his decision to strike Syrian government targets by the reaction to the chemical weapons attack from his daughter Ivanka, who was "heartbroken, and outraged", his son said.
Mr Trump said his father's action on Syria showed he was not tied to Russia, as some critics claim.
"If there was anything that Syria did, it was to validate the fact that there is no Russia tie."
Writing in the Daily Telegraph, Tory former foreign secretary Lord Hague, said the "sad truth" has dawned on President Trump that "Russia under Putin is not a reliable partner".