French Foreign Minister Laurent Fabius said France will not accept delays in the transfer.
"We need quick results," Fabius said.
European Union Foreign Affairs Secretary Catherine Ashton said she supported the French plan to bring the issue to the Security Council, saying the proposal "now needs to be fully worked up as quickly as possible."
Even Sen. John McCain, R-Arizona, a proponent of a military strike on Syria and robust aid to the rebels, said the idea was worth exploring.
"I'm very, very skeptical," he said on CNN's "New Day." "But the fact is, you can't pass up this opportunity -- if it is one."
The Russian proposal surfaced publicly on Monday, when Kerry -- responding to a reporter asking what Syria could do to stop a U.S. attack -- suggested that Syrian President Bashar al-Assad "could turn over every single bit of his chemical weapons to the international community in the next week."
"He isn't about to do it, and it can't be done, obviously," Kerry added.
His spokeswoman, Jen Psaki, sought to roll back the comments, saying the secretary was simply responding to a "hypothetical."
But it turns out Putin brought the idea up to Obama last week, a senior administration official said Monday night.
Kerry and Lavrov have also been discussing ways for Moscow to get involved for more than a year, the official said. But U.S. officials didn't realize how serious Russia was until Lavrov seized on Kerry's comment on Monday, the official said.