"It is the president's responsibility to make his case to the American people and their elected representatives," Steel said in a statement, adding that "all votes authorizing the use of military force are conscience votes for members, and passage will require direct, continuous engagement from the White House."
Obama met Monday with two veteran Republican senators -- John McCain of Arizona and Lindsey Graham of South Carolina -- who emerged to say they could support a more precise and robust strategy than the president initially outlined.
In particular, McCain and Graham said Obama pledged increased military aid to opposition forces in Syria that would bolster their fight against al-Assad at the same time as U.S. military attacks expected to to involve cruise missile strikes on Syrian military command targets.
After Obama met Tuesday with Boehner, Pelosi and the chairs of several national security committees in Congress, legislators from both parties said they expected the initial resolution proposed by the president to be revised to address their concerns.
In particular, they said it would define the mission more narrowly and specify no "boots on the ground," which means no U.S. troops would be deployed to Syria.
West noted that no matter what lobbying takes place, "there are some Republicans who will vote 'no' just because the idea came from President Obama" because "they detest everything he stands for."
Moderates, meanwhile, may face the prospect of a primary challenge from the more extremist wing of their respective party if they authorize a war resolution, he said.
"Anti-war sentiment remains very strong within the Democratic Party," West said, noting that grass-roots activists on the left opposed the Iraq war at the height of post 9/11 patriotic fervor. "The idea of another foreign intervention would be of great concern to those people."
In the end, West said he expects Obama's resolution to win approval because "the president has laid national prestige on the line."
However, a House GOP leadership aide told CNN that "it is going to be a big lift to get this done."
"We're only going to be able to help the president as much as he's willing to help himself," the aide said on condition of not being identified, noting Obama must be personally involved, make the case for military action and "prove that we have a military plan that will work and not drag us into the mud for a long time."
Meanwhile, a Democratic congressman also speaking on condition of not being identified said a "hardcore/progressive" group of Democrats will oppose the resolution, meaning the lobby effort should target "right of center Democrats who focus on national security issues and centrist Democrats."
"It is way too early to predict how the vote is going to go," the Democratic congressman said.