If the US text, which was seen by Reuters, is put to a vote, diplomats said it would likely be vetoed by Syrian ally Russian Federation.
The renewed U.S. push to establish the United Nations Independent Mechanism of Investigation (UNIMI) comes after Russian Federation killed off a previous UN-led probe in November by vetoing the renewal of its mandate.
The council met publicly on Monday over the latest suspected chemical weapons attack.
At the council meeting, Nebenzia warned that a United States military strike against Syria could have "grave repercussions" and stressed that the use of chlorine or sarin had not been confirmed in the attack.
After the defeat of Russia's second resolution, Russian ambassador Vassily Nebenzia said to USA ambassador Nikki Haley: "I would once again ask you, once again beseech you, to refrain from the plans that you're now developing for Syria".
"This is basically a diplomatic set-up", said Richard Gowan, the United Nations expert at the European Council on Foreign Relations. Russian Federation and Syria have said there was no evidence of a chemical weapons attack. "A breakdown at the United Nations will also make it easier for France to justify strikes".
He also discussed with British Prime Minister Theresa May the preparation for a strike on the Syrian government, which it blames for the chemical attack.
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US Ambassador Nikki Haley urged the council to take action, but warned that the United States was ready to respond.
Over 70 people died and a further 500 were injured in the incident in the town of Douma near Damascus on Saturday.
Since the two Russian resolutions did not get a majority, none of the Western permanent members - the US, France or Britain had to veto them.
Russian President Vladimir Putin called the operation an "act of aggression", and Iran said it was a "criminal" attack.
He told the council, "A broad arsenal of methods is being leveraged - slander, insults, hawkish rhetoric, blackmail, sanctions and threats - to use force against a sovereign state".
Even as the Security Council wrangled over a resolution to confer the Council's authority on the probe, the Organisation for the Prohibition of Chemical Weapons (OPCW) announced it was sending a fact-finding mission to Douma, where the chemical weapon attack is said to have taken place. It blamed Islamic State militants for mustard gas use. Moscow slammed the joint United Nations and OPCW inquiry as flawed.
In January, Russia presented its own draft resolution setting up a new panel, but Western powers said Moscow's proposal would give the Syrian government an upper hand over any investigation of attacks on its territory.
In response to Haley's push for a vote on Tuesday, Nebenzia told the council behind closed doors that he, too, could put his draft to a vote.