Washington's European allies gave a similarly cautious welcome to the plan, and UN Secretary General Ban Ki-moon issued his own plea for a mission to secure and dispose of the weapons.
Russian Foreign Minister Sergei Lavrov met his Syrian counterpart and urged Damascus to "place chemical weapons under international control and then to have them destroyed."
Tony Blinken, deputy US national security advisor, said Washington would consult Russia over the initiative but expressed doubt about the trustworthiness of Syria's leadership.
"We would welcome a decision and action by Syria to give up its chemical weapons," he said, but adding that Syria's "track record to date, doesn't give you a lot of confidence."
The day had begun with confusion, when US Secretary of State John Kerry appeared to suggest that Assad could avoid a military strike by agreeing to give up chemical arms.
This led to speculation after that Washington and Lavrov had discussed the plan beforehand, but Kerry's aides denied this, insisting he had no forewarning of the Russian offer.
Nevertheless, Kerry's predecessor Hillary Clinton, speaking after meeting Obama at the White House, linked the two statements.
"If the regime immediately surrendered its stockpiles to international control, as was suggested by Secretary Kerry and the Russians, that would be an important step," she said.
Speaking in Moscow, Syria's Foreign Minster Walid al-Muallem welcomed the Russian move, though it was not immediately clear if a still defiant Assad would give his assent.
"I note that Syria welcomes the Russian initiative based on the Syrian leadership's concern about the lives of our nationals and the security of our country," Muallem told the Russian state news agency ITAR-TASS.
The rebels battling Assad, who saw hope in the United States' threat to bomb the regime, denounced the idea as a plot by Russia's President Vladimir Putin to protect Assad.
Britain's Prime Minister David Cameron also expressed concern that the plan might be "a distraction tactic" but broadly welcomed it.
German Chancellor Angela Merkel described the Kremlin's proposal as "interesting" but added that she hoped it would be put into place quickly and not simply be used to "buy time."
And France, the only Western ally to have offered to take part in a US-led strike, said Assad must commit "without delay" to the elimination of his chemical arsenal.
UN leader Ban, meanwhile, called for the creation of UN supervised zones in Syria where the country's chemical weapons can be destroyed.
He told reporters he may propose the zones to the Security Council if UN inspectors confirm banned weapons were used and to overcome the council's "embarrassing paralysis" over Syria.
For his part, Assad warned in an interview with US television that the United States will "pay the price" if it attacks Syria.
Obama was due to give no less than six US television interviews to defend his strike plan to the American public and lawmakers, before giving a major national address Tuesday.
In the meantime, US cruise missile destroyers are idling in the Eastern Mediterranean, preparing for what American officials described as an extremely limited, precise punitive strike.
Assad's comments did not rise to the level of a precise threat, but will do nothing to calm fears that Syria and it allies Hezbollah and Iran could act to destabilize its neighbors.
A White House spokesman responded that the United States is ready for any move by Assad.
"He has no interest in escalating this conflict, frankly," Ben Rhodes, spokesman for Obama's National Security Council, said of the Syrian leader.
The US Congress returned to work after its summer recess to face a barrage of classified briefings that the White House hopes will convince lawmakers to back a strike.
According to US intelligence, on August 21 a chemical attack against rebel-held suburbs of Damascus killed more than 1,400 people, including 400 children gassed in their beds.
Other outside estimates set a lower but still high death toll, but Western capitals and the Arab League have condemned the alleged barrage as a war crime and blamed it on Assad's regime.
Obama has argued that an international military strike is necessary to defend the long-established international taboo against the use of such weapons.
Nine days ago, in a surprise move, he asked US lawmakers to authorize military action, triggering a bitter debate that could forever undermine his presidential authority.
Polls show a war-weary American public opposes action against Syria and the US lower house, the House of Representatives, is led by Republicans who oppose Obama's every move.
Some liberal anti-war Democrats are also expected to oppose the motion, and the support of pro-war neo-conservatives in the Republican ranks may not be enough to push it through.
British lawmakers have already vetoed their country's involvement in strikes, and Syria's ally Russia has stymied any attempt to win UN legal backing for action.
Obama has refused to rule out acting alone, with neither congressional nor international support, but defeat at home would be a blow to his credibility and strengthen Assad's hand.
Fighting erupted in Syria in March 2011 when Assad's forces launched a brutal crackdown on a popular revolt against his rule, and soon escalated into an all-out civil war. The UN estimates more than 100,000 have died.