The sixth round of UN-backed negotiations is the latest drive to bring a political solution to the conflict which has claimed more than 320,000 lives since 2011.
The talks opened with tensions high over a US charge that Syrian President Bashar al-Assad's government was using a prison crematorium to hide evidence of thousands of murdered detainees.
Syria's government delegation, led by Bashar al-Jaafari, returned to UN headquarters Wednesday afternoon for a meeting with talks mediator Staffan de Mistura.
They were expected to have discussed the UN's proposal to create a "consultative" committee of civil society activists and technocrats tasked with setting a roadmap to a new constitution.
The team would begin work immediately on "specific options for constitutional drafting" in order to "prevent a constitutional or legal vacuum at any point during the political transition process being negotiated," according to a copy of the document seen by AFP.
The opposition High Negotiations Committee (HNC), which has said it has "reservations" about the draft, met with de Mistura later on Wednesday.
- 'No more committees' -
"We don't know what the aim is or how it will be implemented. We don't want there to be more committees outside of our control," the HNC's Munzer Makhos told AFP.
A new constitution is one of four separate "baskets" on the agenda at the talks, alongside governance, elections and combating "terrorism" in the war-ravaged country.
There is meanwhile little hope of a breakthrough, with the HNC continuing to insist on Assad's ousting as part of any political transition -- a demand seen by the government as a non-starter.
And US claims of new regime atrocities at the notorious Saydnaya prison near Damascus have cast additional shadows over the talks.
Satellite images released by Washington appeared to support earlier claims by rights groups that Saydnaya is an execution centre, a claim Damascus rejected.
Relatives of detained Syrians -- some of them imprisoned in Saydnaya -- gathered outside the UN in Geneva Wednesday, carrying portraits of their missing loved ones and tearfully writing their names on long strips of paper.
"Those on the outside think a Syrian detainee is just living in a locked room with a bed and food -- but he's living in a coffin," 42-year-old Amina Kholani told AFP.
The plump woman in a white head scarf was lobbying to free three relatives still held in Saydnaya, and her husband was held there for a year before Syria's uprising started in 2011.
"Sometimes he'd come out unable to walk from the lack of food, or from the torture and the blows," Kholani recalled.
- 'Different role for Russia' -
The HNC has said the accusations about Saydnaya demand an international response, including from Russia.
"For us, we need to see a different role for Russia -- not only helping Assad commit crimes. We need to see Russia help Syrians decide their fate and their destiny," said HNC spokesman Salem al-Meslet.
Longtime regime ally Moscow has become a key player in Syria's crisis, launching an air war in support of Assad in 2015 and brokering several agreements aimed at ending the war.
The latest effort is a tripartite agreement between Russia, fellow Assad ally Iran, and rebel backer Turkey to create four "de-escalation" zones across some of Syria's bloodiest battlegrounds.
It was signed on May 4 in the Kazakh capital Astana as part of parallel negotiations.
Russia's deputy foreign minister Gennady Gatilov met with de Mistura on Wednesday. He also met with Jaafari on Tuesday evening, and was expected to meet with the HNC delegation Thursday.
Syrian peace efforts have also been marked by Washington's all-but withdrawal from the process since President Donald Trump came to power in January.
But on Wednesday, de Mistura met with US Ambassador to International Organisations Henry Ensher to discuss Syria's conflict, according to the UN.