And a recent incident involving the use of chemical weapons and the USA response is also to be discussed.
Tillerson reiterated the US stance that Syrian President Bashir al-Assad was responsible for last week's chemical weapons attack that killed scores of Syrian civilians.
It also said that Trump thanked for the support given by the United Kingdom after the missile strikes against Assad regime.
Even though the Kremlin said last week that its support for Assad isn't unconditional and emphasized that it can't control the Syrian leader's every move, such statements sound more like rhetoric aimed at deflecting Western criticism than a signal of Moscow's readiness to bargain over Assad's fate. "Our view is the reign of the Assad family is coming to an end".
"Right now we're not getting along with Russian Federation at all", Trump said flatly during a White House news conference with NATO Secretary-General Jens Soltenberg.
Russian Federation has rejected accusations that Assad used chemical arms against his own people and has said it will not cut its ties with Assad, who has been locked in a six-year-old civil war that has devastated Syria and displaced half its population.
The meeting comes amid a fresh war of words over who is responsible for last week's chemical weapons attack in Syria. "But we're going to see what happens", Trump told the news conference.
Tillerson spoke after a meeting of the "likeminded" countries was hastily arranged on the sidelines of the summit of the Group of Seven industrialized economies in Italy.
"We want to relieve the suffering of the Syrian people".
"We both have agreed our lines of communication shall always remain open", Tillerson said.
Russia's veto of a United Nations resolution condemning a deadly chemical attack in Syria and pushing Bashar Assad's regime to co-operate with an global inquiry put the country on the "wrong side" of the argument, the Foreign Secretary said.
Only weeks ago, it appeared that Trump, who praised Putin throughout the US election campaign, was poised for a potentially historic rapprochement with Russian Federation.
"Russia's allegations fit with a pattern of deflecting blame from the regime and attempting to undermine the credibility of its opponents", the report read.More news: Fiji win Hong Kong Sevens in style
His comments came moments before President Donald Trump said relations with Russian Federation had reached a "all-time low".
Speaking of the missile strikes he authorized against the base the carried out the chemical attack, Trump later added: "I have absolutely no doubt we did the right thing".
On Monday, Col. John J. Thomas, a US military spokesman, said the USA has taken extra defensive precautions in Syria in case of possible retaliation against American forces for the cruise missile attack. "It spins out multiple, conflicting accounts in order to create confusion and sow doubt within the global community".
Russian Federation has argued that civilians in Khan Sheikhoun were exposed to toxic agents from a militants' arsenal hit by a Syrian air strike.
Damascus, and its Russian backers, deny any chemical weapons were used.
The Russian Defense Ministry said Wednesday that those strikes were not as effective as the USA claims.
"A successful ouster of a dictator is for me very hard to remember", Lavrov said.
He also rejected American claims of incontrovertible evidence that Assad ordered the chemical attack. Now, the Trump administration has to back up that message with a coordinated, coherent foreign policy that addresses not only Syria and Russian Federation but other nations that might still be unclear about the White House's willingness to become involved in world affairs. He likened the USA response in bombing a Syrian air base to the 2003 invasion of Iraq, arguing that now just as then Washington relied on false data about chemical weapons.
The Pentagon has said it is sure Assad unleashed a chemical attack on April 4 that killed at least 87 civilians in the rebel-held Syrian town of Khan Sheikhun.
Health Minister Recep Akdag said traces of sarin gas was found in the blood and urine of victims, according to the state-run Anadolu news agency.
Before the meeting started, the foreign minister of Iran, which supports Assad, asked to speak to Italian Foreign Minister Angelino Alfano to discuss Syria, Italian diplomats said.
Syrian government forces battling insurgents in Daraa have dropped several barrel bombs on the southern city, opposition activists said Tuesday, a day after the White House suggested that the use of the crude explosive devices could provoke a USA response.
"Syria is not the only issue here", Kremenyuk said.