Russian Federation has vetoed a UN Security Council resolution demanding a swift investigation into a chemical weapons attack in Syria. Mr Putin had even granted Mr Tillerson a friendship honour.
Speaking alongside Mr Tillerson at a news conference, Mr Lavrov said relations between the two nations "are not the most calm" and some issues were "time bombs" inherited from the Obama administration.
Tillerson and Lavrov maintained a diplomatic tone during their chilly joint press conference, but the growing tensions following the US strike on a Syrian airbase last week in retaliation for Assad's lethal chemical attack on his own people were crystal-clear, as Tillerson suggested war crimes charges could be levied against the Syrian leader and repeated the White House's belief that "the reign of the Assad family is coming to an end".
Maynes adds, "The Kremlin argues Assad gave up his chemical weapons stores under a 2013 agreement Moscow brokered with the Obama administration; Tillerson has accused Russian Federation of "incompetence" in failing to enforce that deal". But it appears they didn't discuss additional sanctions - until they were in front of the press.
But Tillerson spokesman R.C. Hammond says "no agreement was reached".
The press conference came just moments before Russian Federation again cast a veto at the UN Security Council, blocking a bid from the US, UK and France to condemn the suspected gas attack and push the Syrian government to cooperate with investigators.
Putin, meanwhile, accused Assad's opponents of planning to stage chemical attacks to be blamed on Damascus in order to lure the United States deeper into the conflict.
Slate looks at a couple reasons for why President Trump hasn't been almost as pro-Russia as candidate Trump, including the possibility that he's been prevented from following pro-Russia policies by political realities in Washington, or simply that he's changed his mind on Russian Federation.
"Right now we're not getting along with Russian Federation at all", Mr Trump said flatly during a White House news conference with Nato Secretary-General Jens Stoltenberg.
The diplomats know each other well from Mr Tillerson's days as Exxon Mobil chief executive.
Secretary Tillerson: The final outcome does not provide for a role for Assad in the future governance of #Syria.
"I expressed the view that the current state of U.S. -Russia relations is at a low point.", Tillerson said. "He hasn't threatened us at all", Lavrov responded. "But we're going to see what happens". Both Washington and Moscow say their main enemy is Islamic State, although they back opposing sides in the wider civil war which has killed more than 400,000 people and spawned the world's worst refugee crisis.
"And of course, that's not to mention that apart from the statements, we observed very recently the extremely worrying actions, when an illegal attack against Syria was undertaken".
Secretary Tillerson comments on the importance of cyber security discussions with #Russia.
"For the U.S.to change that, Washington would have to field a formidable military challenge to (al-Assad)", White said.
Still, Mr Tillerson sought to stress the positives from his meetings.
He said working groups would be established to improve US-Russian ties and identify problems.