Rep. Mike Rogers, chairman of the House Intelligence Committee, said the U.S. is "being led by the nose by" Russian President Vladimir Putin.
"So, if we wanted a transition with Assad, we just fired our last round, and we have taken our ability to negotiate a settlement from the White House, and we've sent it with Russia to the United Nations," Rogers said. "That's a dangerous place for us to be if you want an overall settlement to the problems."
Russia, which already has rejected three Security Council resolutions on Syria, would be sure to veto a U.N. move toward military action, and U.S. officials said they did not contemplate seeking such an authorization.
Obama said Saturday that "if diplomacy fails, the United States remains prepared to act," and Secretary of State John Kerry warned during a visit to Israel on Sunday that "the threat of force is real" if Assad fails to live up to the terms of the agreement.
Sen. Bob Corker of Tennessee, the top Republican on the Senate Foreign Relations Committee, said the threat of force "is still very much in Russian hands."
"That's the most important element, is the veto piece," Corker said. "So in many ways, our credibility in the region, and certainly relative to the chemical warfare, is very much driven by Russia, which has its hands firmly on the steering wheel. "
Republican Sens. John McCain of Arizona and Lindsey Graham of South Carolina, who are among Obama's sharpest foreign policy critics and support greater U.S. assistance for Syria's rebels, said the agreement will embolden enemies such as Iran.
Democrats insisted that while the agreement itself doesn't commit the U.S. to using force, the option of acting independently of the U.N. remains.
Democratic Sen. Carl Levin, chairman of the Senate Armed Services Committee, said Russia's primary aim has been to force the U.S. to give up that option. "Russia has failed in that goal," Levin said.
To Democratic Sen. Robert Menendez, chairman of the Senate Foreign Relations Committee, the threat of American military action is "the only reason we've gotten to this point, even to this possibility."
Obama said in an interview with ABC television's "This Week" that if Syria can be stopped from using chemical weapons, "then we may also have a foundation" to begin the process of reaching a political settlement to the civil war.
The president's interview aired Sunday but was taped Friday, before the chemical weapons deal was reached but while Secretary of State John Kerry was engaged in intense talks with Russian Foreign Minister Sergei Lavrov in Geneva, Switzerland.
Obama said Putin is "protecting" Assad and doesn't share American "values" in Syria. "He has a different attitude about the Assad regime," Obama said. "But what I've also said to him directly is that we both have an interest in preventing chaos, we both have an interest in preventing terrorism. The situation in Syria right now is untenable. As long as Mr. Assad's in power, there is going be some sort of conflict there."
The U.S. says intelligence reports have placed the blame on the Assad government for the attack last month.
But polls showed relatively little support among Americans for a military strike against Syria, even after the Obama administration's efforts to argue that punishing the Assad government for violating international norms of warfare was in the security interests of the U.S.
Obama ordered preparations for American airstrikes, but he decided instead on Aug. 31 to ask for authorization from Congress for military action, only to face growing skepticism from both Republican and Democratic lawmakers. Then came the Russian proposal for international control of Syria's chemical weapons, and Obama asked Congress, already largely opposed to military intervention, to delay a vote.
The deal to destroy Syria's chemical weapons also offers the potential for reviving international peace talks to end a civil war that has claimed more than 100,000 lives and sent 2 million refugees fleeing for safety, and now threatens the stability of the entire Mideast.
Rogers spoke on CNN's "State of the Union." Corker and Levin were on CBS' "Face the Nation," and Menendez appeared on NBC's "Meet the Press."