The resolution, if it passes, would blacklist 11 Syrian military commanders and officials.
The draft resolution also seeks to ban the sale or supply of helicopters to the Syrian government and to blacklist 10 government and related entities involved in the development and production of chemical weapons and the missiles to deliver them.
It calls for an asset freeze and travel ban for the individuals and entities across all UN member states.
Human Rights Watch (HRW) said in a recently released major report that Syrian government forces had carried out at least eight chemical attacks in late 2016 as they were fighting to capture second city Aleppo.
The report followed the discovery by a joint inquiry of the United Nations and the Organization for the Prohibition of Chemical Weapons (OPCW) that Syrian government forces were responsible for three chlorine gas attacks and that Islamic State jihadists had used mustard gas.
Syria continuously denies using chemical weapons during the civil war, claiming that the allegations are "a campaign of lies" made by “Western circles”.
Chlorine use as a weapon is banned under the Chemical Weapons Convention, which Syria joined in 2013 under pressure from Russia.
The Security Council diplomat told Reuters the draft resolution would be brought to a vote next week unless a "really compelling argument" against it emerged from the talks.
The draft resolution is a French and British initiative and would also be supported by the United States, the diplomat said on condition of anonymity. However, it is likely to be vetoed by Russia, a close ally of Syria which has already used its veto power six times to block action on Syria.
The U.S. and Russian delegations to the United Nations did not immediately respond to a request for comment.
Following the UN-OPCW investigation, the United States in January blacklisted 18 senior Syrian officials it said were connected to the country's weapons of mass destruction program.
At least nine of the 11 officials in the UN draft resolution were also blacklisted by the U.S. government, according to Reuters.