Russian President Vladimir Putin is to hold rare talks with US President Barack Obama to outline his proposals.
Moscow has suggested there are plans to form an international contact group on Syria that will include Russia, the US, Iran, Turkey, Saudi Arabia and Egypt.
The Russians are a key ally of Syrian President Bashar al-Assad.
Western leaders have recently softened their stance towards him – conceding that he might be able to stay on during a political transition.
The threat of Islamic State (IS) extremists and the flow of Syrian refugees to Europe has added urgency to the search for a deal to end the civil war.
Russian deputy foreign minister Mikhail Bogdanov suggested both Russia and the United States would take part in peace talks on Syria next month, along with Egypt, Iran, Saudi Arabia and Turkey, the RIA Novosti news agency reported.
Mr Putin has reiterated his support for President Assad, who Western countries and the Syrian opposition have said must go.
Mr Putin, who has strongly reinforced Russia’s military presence in Syria, has called for a regional “co-ordinating structure” against IS, and said the Syrian president’s troops were “the only legitimate conventional army there”.
He said Russia would not participate in any troop operations in Syria.
Relations between Russia and the West have been strained over Moscow’s annexation of Ukraine’s Crimea peninsula last year and its support for separatist rebels in Ukraine’s east.
Mr Putin will also meet Iranian President Hassan Rouhani and Cuban President Raul Castro on the sidelines of the assembly, the Kremlin was quoted as saying by Reuters.
President Rouhani – a key regional ally of President Assad – says the government in Damascus “can’t be weakened” if IS militants are to be defeated.
US Secretary of State John Kerry, however, said the efforts were “not yet co-ordinated” and the US had “concerns about how we are going to go forward”.
UK Prime Minister David Cameron is expected to reflect the softening of the Western stance against Mr Assad this week.
He is set to tell leaders at the gathering in New York that Mr Assad could remain temporarily in power at the head of a transitional government.
Mr Cameron – along with Mr Obama and French President Francois Hollande – has previously demanded that Mr Assad be removed from power as a condition of any peace deal, a position consistently rejected by Mr Putin.
In addition to Russia’s military build-up in Syria, Iraq on Sunday announced that it had signed an agreement on security and intelligence co-operation with Russia, Iran and Syria to help combat IS.
And France said on Sunday it had carried out its first air strikes against IS in Syria, destroying a training camp.
A US-led coalition has been carrying out air strikes against IS in Syria and Iraq for more than a year. France, like the UK, has previously confined its air strikes against the Islamic State group to Iraqi airspace.
The UK announced this month it had carried out a drone strike against two British citizens in Syria, but has yet to fly manned operations in Syrian airspace.
More than 200,000 Syrians have been killed since the country erupted into civil war in 2011, and Islamic State militants took control of swathes of the country in 2014. Mr Assad has been accused of killing tens of thousands of his own citizens with indiscriminate bombing in rebel-held areas.
Approximately four million Syrians have fled abroad so far – the vast majority are in neighbouring Turkey, Lebanon and Jordan – and more are on the move.