"The world's two foremost nuclear powers can not have this kind of relationship", Tillerson said at a joint press conference with Russian Foreign Minister Sergey Lavrov following a two-hour meeting with Putin.
Russian Federation has vetoed a Western-backed draft UN Security Council resolution on last week's suspected chemical weapons attack in Syria.
Earlier in the day, Lavrov accused the USA of conducting an unlawful attack against Assad's forces.
At one point, Lavrov also pushed back at the US intelligence community, which has reason to believe that Russian Federation interfered in USA elections.
This occurs as tensions between the two countries continued to escalate over the American military strike in Syria, Russian Spokesman Dmitry Peskov confirmed. He also noted they wanted a "frank investigation" into the Syrian chemical attack, believed to be launched by President Bashar al-Assad.
The fallout over the chemical attack follows comments by the Trump administration and Russian Federation that a reset in relations between the countries was possible after decades of hostility.
The Moscow news conference came after Putin met Tillerson for the first time since Trump took office.
But Mr Trump also said: "Just so you understand, we're not going into Syria".
Russian Federation has vetoed a United Nations resolution condemning the reported use of chemical weapons in Syria and urging a speedy investigation.
Rex Tillerson, Secretary of State: "Our priority in Syria, John, really hasn't changed".
Russian President Vladimir Putin said on Wednesday (April 12) that trust had eroded between the United States and Russia under President Donald Trump as Moscow delivered an unusually hostile reception to Secretary of State Rex Tillerson in a face-off over Syria.
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Spokeswoman Megan McCarthy said Wednesday that the passengers can take the compensation in cash, travel credits or miles . Video shot by passengers showing the Dao's bloodied face went viral on social media, prompting a storm of protest.
Putin is not expected to meet with Tillerson while he is in Moscow.
The talks come after Lavrov issued Tillerson a warning Wednesday against any further United States strikes on the Syrian regime.
In what was effectively an ultimatum, Tillerson on Tuesday said that Moscow must calculate the costs of remaining an ally of Assad, the Iranians and Lebanon's Shiite militia Hezbollah.
US Defence Secretary James Mattis said Washington has "no doubt" that Assad was behind the massacre.
The proposed resolution would have condemned the alleged attack and expressed the council's full backing to investigators from the Organization for the Prohibition of Chemical Weapons (OPCW).
Subtly mocking his guest, Lavrov said their talks were especially important because "not all key positions in the State Department have been filled yet".
Syria has begun reconstruction efforts at the airbase destroyed by US cruise missiles Thursday, CBSNews reports.
USA relations with the Kremlin have become politically toxic for the White House on the back of claims Putin conspired to get Trump elected.
"Our big mission is getting rid of ISIS (ISIL)", Trump said.
"We're not going into Syria", Trump told Fox Business News reporter Maria Bartiromo in an interview set to air Wednesday. I think he'd like me.
Tense comments and warnings from Russian Federation set the tone Wednesday for Secretary of State Rex Tillerson as he attempts to persuade Moscow to abandon its support for Syrian leader Bashar Assad.
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