But with U.S. officials suggesting Russian forces may have colluded in the latest atrocity blamed on President Bashar al-Assad's regime, it is not clear whether Tillerson will be invited to meet President Vladimir Putin, with the Kremlin only saying there was a "probability" they could talk.
Lavrov blasted US claims that it has "irrefutable evidence" of election interference.
Tillerson on Wednesday acknowledged that the US and Russian Federation had "sharp differences", but called for cooperation between the two countries. Both Washington and Moscow, Tillerson said, want a "unified and stable" Syria, so the country does not become a "safe haven" for terrorists. During his presidential election campaign, Trump said he would like the United States to have better relations with Russian Federation.
On Tuesday he said that "fake chemical attacks" were being prepared in other regions, created to be blamed on the Syrian government.
Still, Mr Tillerson sought to stress the positives from his meetings. The diplomats know each other well from Tillerson's days as Exxon Mobil CEO.
It was unclear whether Putin, who once gave Tillerson an "Order of Friendship" award, would grant the visiting American an audience in Moscow.
Tillerson met with Putin and Lavrov on Wednesday. Jabbing at USA credibility, the Russian leader reminded reporters about unfounded US claims of Iraqi weapons of mass destruction, used to justify America's 2003 invasion.
Russian Federation has defended the Syrian government, a staunch ally, against United States claims it was behind an alleged chemical attack in Syria's Idlib province last week in which scores died.
But Tillerson's claim is one President Barack Obama, too, argued for years, only to see Assad outlast his own term in office. Putin has spoken of having proof that it was a "false flag" operation meant to rally global opposition to Assad.
Earlier Russian Foreign Minister Sergey Lavrov said after talks with his U.S. counterpart in Moscow that if there is not an worldwide inquiry then it will mean that the global community is not interested in establishing the truth.
US officials on Tuesday laid out the case against Assad over the chemical attack in the Idlib province that prompted Trump to order an airstrike with 59 cruise missiles on a Syrian airbase last week.
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The military action, coupled with American officials' change of tone over Assad's future, stirred speculation that the United States and its Western allies were ready to place boots on the ground in Syria.
"There is a low level of trust between our two countries", Tillerson said candidly. It denied preferring Trump to his Democrat rival, Hillary Clinton, during the election campaign and said it would judge his administration by its actions rather than its words.
A fact-finding mission from the Organisation for the Prohibition of Chemical Weapons (OPCW) is investigating the attack.
Syrian U.N. Ambassador Bashar Ja'afari said Syria had sent dozens of letters to the Security Council, some detailing "the smuggling of sarin from Libya through Turkey on a civilian air plane by using a Syrian citizen".
"We're not going into Syria".
"We discussed Assad today", said Lavrov.
Tillerson entered Wednesday's talks facing stiff opposition to his demand that Russian Federation end its support of the Syrian regime following the deadly April 4 chemical weapons attack in the town of Khan Shaykhun.
Only weeks ago, it appeared that Trump, who lavishly praised Russian President Vladimir Putin throughout the campaign, was poised for a potentially historic rapprochement with Russia. Any expectations of an easy rapport have crashed into reality amid the nasty back-and-forth over Syria and ongoing USA investigations into Russia's activity connected to the US presidential election.
"Secondly, I believe that Russian Federation wants to be a respected global partner". "I will be frank and say that we have a lot of questions regarding the very ambiguous and contradictory ideas on a whole range of bilateral and global issues".
He also complained about the mixed messages coming out of Washington on the Trump administration's policy on Syria, with the US envoy to the United Nations, Nikki Haley, making clear Assad should have no future in Syria as Tillerson took a softer line.
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