If it can not support a future Security Council resolution to that end, it should abstain, Tillerson said.
The Russian proposal focuses on collecting information about "non-state actors" - including extremist groups - that may be involved in using chemical weapons in Syria or elsewhere, rather than on possible use by the Assad government. Media reports said they were fired from the Syrian Kurdish enclave of Afrin.
Syria's plight actually got a bit worse this week, as Turkey invaded the border region known as Afrin.
After intense exchanges, Turkey's forces took control of the hill of Barsaya, a key strategic point in the Afrin region.
Addressing Turkish complaints that the United States has not kept to its promises to take back weapons supplied to the Syrian Kurdish fighters, the official said Washington meant to "fulfil the commitment" but would not provide a time frame.
Nebenzia says he is circulating a resolution proposing the establishment of a new worldwide investigative body to identify those responsible for chemical attacks.
The United Nations says the risk to the area's 324,000 civilians is high.
The United States has around 2,000 special forces troops in Syria, officially as part of an worldwide USA -led coalition, assisting the Kurds in battle against Islamic State.
"We are determined Afrin will be sorted out, we will take no step back", Mr. Erdogan said in a televised speech on Monday, also claiming that Turkey reached an agreement with Russian Federation regarding the offensive.
"So we urge Turkey to exercise restraint in the military actions and the rhetoric and ensure that its operations are limited in scope and duration", he added.
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The U.N. says the displaced fled from areas near the border but are still inside the enclave.
The U.S. policy is still to have Mr.al-Assad removed as Syria's leader.
The Kurdish-led administration of north-eastern Syria appealed for a mass mobilisation in defence of Afrin.
A separate official said Trump planned to speak with Erodgan on Wednesday by phone.
It was not immediately clear which issues would be addressed but Turkish-American relations have been strained.
Turkish military's operations in Syria will continue "until all terrorists are fully eliminated" and 3.5 million Syrian refugees return home, President Recep Tayyip Erdogan's spokesman said on Tuesday.
The operation has opened a new front in Syria's multisided civil war and could threaten USA plans to stabilise and rebuild a large area of northeast Syria - beyond President Bashar Assad's control - where the United States helped a force dominated by the YPG to drive out the Daesh terror group fighters.
Hiddo said since 2012, when Afrin came under the control of the YPG, the town has suffered varying degrees of siege, first by opposition fighters, then IS, and now at the hands of the Turkish-led forces.
The French government is freezing assets of companies that help furnish material to a Syrian lab accused of producing chemical weapons.