WASHINGTON — President Obama woke up Monday facing a Congressional defeat that many in both parties believed could hobble his presidency. And by the end of the day, he found himself in the odd position of relying on his Russian counterpart, Vladimir V. Putin, of all people, to bail him out.

The surprise Russian proposal to defuse the American confrontation with Syria made a tenuous situation even more volatile for a president struggling to convince a deeply skeptical public of the need for the United States to respond militarily in yet another Middle Eastern country, this time in retaliation for the use of chemical weapons. It could make the situation even more precarious. Or it could give Mr. Obama an escape from a predicament partly of his own making.

In effect, Mr. Obama is now caught between trying to work out a deal with Mr. Putin, with whom he has been feuding lately, or trying to win over Republicans in the House who have made it their mission to block his agenda. Even if he does not trust Mr. Putin, Mr. Obama will have to decide whether to treat the Russian proposal seriously or assume it is merely a means of obstructing an American military strike.

“Putin knows that everyone wants an out, so he’s providing one,” said Fiona Hill, a former national intelligence officer and co-author of “Mr. Putin: Operative in the Kremlin.” “It seems like a bold idea that will get everyone, including Obama, out of a bind that they don’t want to be in.”