The Russian offer provides Democrats an opportunity to dodge a politically dangerous vote while avoiding a damaging loss for Obama if Congress failed to approve the resolution.
Americans are overwhelmingly opposed to a U.S. strike against Syria, polls show.
"I think what you need to do is make sure the president has the opportunity to speak to all 100 senators and all 300 million American people before we do this," Reid said.
Reid's move, aides said, came as a result of the Russian offer and not because votes in the Senate are lacking.
Reid made the claim Monday that he had secured the 60 votes needed to move forward on the resolution, but only about two dozen Senators have voiced their support.
Opposition to the measure has been increasing in the Senate, even among Democrats, despite intense lobbying efforts by White House officials and President Obama to build support for a strike.
The president is scheduled to make a rare visit to Capitol Hill Tuesday to meet separately with Senate Republicans and Senate Democrats. He is scheduled to address the nation later Tuesday.
Lawmakers Monday night seized on the news of the Russian offer and Obama's comments on a round of television interviews that it could lead to a "breakthrough" in the standoff with Syria over its use of chemical weapons.
During a classified briefing held by White House officials in the Capitol, lawmakers raised the Russian proposal as an alternative route to ridding the Syria of chemical weapons.
"Some of us feel this is the right way to go," Rep. Devin Nunes, R-Calif., said as he left the meeting. "I feel like we have not exhausted all the diplomatic solutions. Look, I don't believe the Russians for one second, but it's better than lobbing a few Tomahawk missiles and calling it a day."
Support for the strike in the GOP-led House is even weaker than in the Senate and lawmakers have not even scheduled a vote or drafted a resolution. Instead, House Republicans have been waiting to see how the resolution plays out in the Democratically run Senate.
On Monday, Democratic backers came up with some unique arguments to justify their support for the strike.
Reid compared the chemical gas attacks in Syria to the gas chambers of Nazi concentration camps.
"I know the options are not good," Durbin said. "They never are in these circumstances. But I also know if we turn our backs on this situation, there will be some dictator in Iran or North Korea who will be emboldened to do even more, to use not just chemical weapons but nuclear weapons."
Rep. Keith Ellison, D-Minn., was among many Democrats crediting President Obama for the Russian offer.
"I don't think the Russia proposal has been made because they got religion," Ellison said. "I think they realize that their client is about to have some serious consequences, and so they have stepped up."
Ellison said the Russian proposal shouldn't cause the U.S. to significantly delay making a decision on Syria, "because there is a fair chance that is exactly why they made this proposal."
Congressional Correspondent Sean Lengell contributed to this report.