Italy, which hosted the G7 gathering in Tuscany, said the idea did not win broad support.
A senior US official says the United States has concluded that Russian Federation knew in advance of Syria's chemical weapons attack last week. "I welcome this renewed harmony".
The U.S. -backed Syrian Democratic Forces has been on the offensive since November capturing territory with the aim of laying siege to Raqqa.
Before heading to Russia, Tillerson told reporters that Moscow had either failed to take seriously its obligation to rid Syria of chemical weapons, or had been incompetent.
US Treasury Secretary Steven Mnuchin said last week that the USA would soon impose additional sanctions on Assad, as part of a military, diplomatic and financial response to a chemical attack blamed on the regime.
"We are calling on Russian Federation and Iran to do everything possible to bring about this settlement and to work with the global community to ensure last week's shocking events aren't repeated".
Alfano said the G7 did not want to put Russian Federation in a corner, but rather sought a constructive relationship with Moscow. Tillerson headed to Moscow later that day for the start of a series of what promise to be some tough conversations.
He is accusing Syrian rebels of staging the chemical weapons attack.
It's a clear mission, McConnell said.
German Foreign Minister Sigmar Gabriel said Russian Federation, and Assad ally Iran, must be involved in any peace process to end Syria's six-year civil war.
He called Moscow "incompetent" for failing to hold up its end of a deal to rid Syria of chemical weapons, and accused Russian Federation of interfering in European-but not American-elections.
"We have information from various sources that such provocations - and I can not call them anything else - are being prepared in other regions of Syria, including in the southern suburbs of Damascus, where they intend to plant some substance and blame the official Syrian authorities for its use", Putin said.
U.S. Secretary of State Rex Tillerson steps off his plane at Pisa Military Airport in Pisa, Italy, on Sunday, April 9, 2017. He said the United Nations should first investigate the attack. The Syrian government has denied it was behind the assault.
USA officials previously have said Russians routinely work with Syrians at the Shayrat air base where the attack is supposed to have originated.
Since ordering the strike on the Shayrat airbase in western Syria, the U.S. has been sending mixed signals.
Days after the incident, the USA blasted 59 Tomahawk missiles at the Shayrat air base near Homs which Washington believes Assad's jets launched the attack from.