"I expressed the view that the current state of U.S. -Russia relations is at a low point", Tillerson said during a joint press conference with his Russian counterpart, Foreign Minister Sergey Lavrov.
He said U.S. Secretary of State Rex Tillerson is considering the request. Tillerson called for Assad to eventually relinquish power.
U.S. military officials said they observed a drone flying over a hospital in Khan Sheikoun that was providing treatment for gas victims.
Many watching the Trump-Russia storyline online agreed.
At their bilateral press conference, Tillerson left open the possibility that the US might move to charge Assad with war crimes, although he said sufficient evidence to formally level such an accusation had not yet been amassed.
"It can be said that the level of trust at the working level, especially at the military level, has not become better but most likely has degraded", Putin said in an interview broadcast Wednesday by state television channel Mir.
The two leaders were expected to discuss disagreement about whether Syrian President Bashar Assad or his opposition was responsible for a chemical weapons attack last week that led to US retaliatory cruise missiles.
Later Wednesday, Russia blocked the U.N. Security Council's draft resolution condemning Syria and calling for it to comply with an investigation. "And we suspect that they have help", a U.S. military official told reporters at a background briefing Friday.
Still, Tillerson was greeted frostily in the Russian capital as Lavrov began their meeting Wednesday by demanding to know America's "real intentions".
One of Mr Lavrov's deputies was even more scathing.
Combination of file photos show Russian President Vladimir Putin (L) at the Kremlin in Moscow, Russia, Dec. 29, 2016 and U.S. President DonaldTrump (R) at a press conference at the White House in Washington D.C., the United States, on February 15, 2017.
A misdirected airstrike has killed 18 allied fighters battling the Islamic State group in northern Syria, the U.S. military says.
Lavrov said Moscow was hoping to understand Washington's "real intentions" and warned that Moscow considered it "fundamentally important" to prevent more "unlawful" USA strikes against its ally Syria.
Bashar Assad Says Accusations of a Chemical Weapons Attack Are a 'Fabrication'
The US strike on the Syrian air base was the first time Washington has deliberately and directly targeted the Syrian government. US Ambassador to the UN Nikki Haley said the veto left Moscow, a key ally of the Syrian government, with "a lot to prove ".
The palpable tension hanging over Tillerson's trip spoke to a widening chasm between the two world powers.
"The world's two foremost nuclear powers can not have this kind of relationship", he said at a press conference afterwards.
In another possible setback to a thaw with Moscow, Trump said on Wednesday that North Atlantic Treaty Organisation is not obsolete, as he had declared during the election campaign previous year. Interestingly enough, in the same press conference where Trump said relations were poor, NATO Secretary General Jens Stoltenberg addressed the comparison.
Trump said USA relations with Russian Federation were not going well. Right now we're not getting along with Russian Federation at all.
Relations between the United States and Russian Federation appear to have reached depths not seen since the end of the Cold War. "We'll see what happens".
Of Assad, Trump added: "This is an animal".
"Are we insisting on it? No".
A senior administration official who briefed reporters Wednesday said it is "very important for us to call out Russian Federation over their sponsorship of the Syrian regime".
At the time of the incident Syria denied any use of chemical weapons, and Moscow said the deaths had been the result of a conventional strike hitting a rebel arms depot containing "toxic substances".
The strikes came after a chemical attack killed 89 people in the opposition-held Idlib Governorate.
The unscheduled talks between Putin and the secretary came as Moscow warned against the US repeating its missile strike on a Syrian air base.
And as the men entered a day of tense talks, Putin admitted that relations between Washington and Moscow have worsened in the three months that Trump has been in office.