America's stance came as the United Nations yesterday confirmed it had received documents from Syria on joining the Chemical Weapons Convention, which outlaws their production and use.
Earlier, in an interview with Russian state news channel, Rossiya 24, President Assad said his government had agreed to surrender its chemical weapons in response to Russia's initiative and not because of the US threat of attack.
He said Syria will start submitting data on its chemical weapons stockpile one month after signing the convention banning such weapons. Mr Assad said this was the standard process and his country will follow it.
However, speaking in Geneva following the first day of talks with Russian Foreign Minister Sergey Lavrov, US Secretary of State John Kerry said there was "nothing standard about this process" since the Syrian president had already used chemical weapons against his own people.
Mr Kerry said: "The words of the Syrian regime, in our judgment, are simply not enough."
Mr Assad denies that the Syrian government is responsible for a chemical attack on civilians near Damascus which killed thousands on August 21. He blames rebel forces for secretly releasing the gas in a bid to provoke international support for its cause.
Mr Kerry cautioned that a US military strike could occur if Mr Assad does not agree to dismantle his chemical arsenal properly. He said: "There ought to be consequences if it doesn't take place."
His Russian counterpart, Mr Lavrov, said the dismantling of chemical weapons "will make unnecessary any strike against the Syrian Arab Republic".
Mr Kerry and a team of US experts will have at least two days of meetings with their Russian counterparts, although discussions could run over into tomorrow. They hope to emerge with an outline of how some 1000 tons of chemical weapon stocks and precursor materials as well as potential delivery systems can be safely inventoried and isolated under international control in an active war zone and then destroyed.
One official said the task is "doable but difficult and complicated". The official said the US is looking for signs of Russian seriousness and thinks it will know in a relatively short time if Moscow is trying to stall.
Another official described the ideas that the Russians have presented so far as "an opening position" that needs a lot of work and input from technical experts. The US team includes officials who worked on inspection and removal of unconventional weapons from Libya after 2003 and in Iraq after the first Gulf War.
Meanwhile, it was confirmed the CIA has been delivering light machine guns and other small arms to Syrian rebels for several weeks, following President Barack Obama's decision to arm the rebels. The agency has also arranged for the Syrian opposition to receive anti-tank weaponry such as rocket-propelled grenades through a third party.
The US is hoping an acceptable agreement with the Russians can be part of a binding UN Security Council resolution currently being negotiated which would hold Syria accountable for using chemical weapons. Russia, however, has long opposed UN action on Syria, having vetoed three earlier resolution, and it has not indicated it is willing to go along with one now.