Meanwhile on Thursday U.S. President Donald Trump said that "things will work out fine" between the United States and Russian Federation, only a day after he said U.S.
France's U.N. Ambassador Francois Delattre said "with this attack, the Syrian regime has plumbed new depths of horror".
"The world's two foremost nuclear powers can not have this type of relationship", Mr Tillerson warned.
Tillerson and Lavrov are meeting in Moscow amid rising tensions over Syria.
However, the Syrian nerve gas attack blamed on Russia's ally, Syrian President Bashar Assad, and the retaliatory U.S. missile attack against Syria has sent U.S. -Russia ties to rock bottom. Before their face-to-face, the foreign minister warned the United States against carrying out an air strike on Syria similar to the one launched last week.
This view is a change from his stance just two weeks ago, when he said removing Assad was not a priority.
Mr Lavrov also noted that many key State Department posts remain vacant since the new administration took office - a point of sensitivity in Washington.
Secretary of State Rex Tillerson made it clear on Wednesday that the United States will not support the continued rule of Syrian President Bashar al-Assad.
Safronkov, who demanded Rycroft look at him while he was speaking, responded: "I can not accept that you insult Russian Federation". "We'll see what happens".
"Things will work out fine between the United States of America and Russian Federation".
The tense back-and-forth over last week's deadly chemical attack in Syria has added to the strain.
Russian Federation has slammed Washington's attack on a Syrian airbase and, as Mr Tillerson met Mr Lavrov, Putin admitted that relations between Washington and Moscow have worsened in the three months that Trump has been in office. "I do not know who saw them".
"There hasn't been a single fact, although under the pressure of President Donald Trump's foes the White House has been forced to periodically make statements containing unfounded accusations against us".
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".