"Part of it is, are you going to be able to get, say, McConnell to step aside while 40 senators from both sides work out a bargain?" Ornstein said of Senate Minority Leader Mitch McConnell. Persuading House Speaker John Boehner to call a vote on a deal unpopular with his caucus -- as any bipartisan agreement is sure to be -- is another challenge.
Obama is already planning to head out across the country to rally the public behind his insistence that a deal include both higher taxes on the wealthy and spending cuts, The New York Times reported on Monday. The president also plans to maintain his grassroots organization to leverage public support, according to The Times. Last cycle, vestiges of Obama's 2008 field organization were incorporated into the Democratic National Committee to further Democratic policy goals.
The White House, and the president himself, have long touted the benefits of playing an outside game. Senior adviser David Plouffe told reporters last week that Obama's supporters "play a very, very important role" in policy change, citing health reform and student loans as two examples. Obama believes "you're not going to be able to change Washington solely from the inside," Plouffe said.
It's hard to prove, however, that pressure applied by Obama's supporters managed to move legislators during his first term. As the president approaches his second, the pitched battle over deficit reduction may be a particularly difficult space for him to leverage his grassroots organization.
"To mobilize people out there, there needs to be a kind of clarity," said Marshall Ganz, senior lecturer in public policy at Harvard's John F. Kennedy School of Government. It's extremely hard to mobilize people behind a plan that doesn't yet exist, or is in the process of being tweaked and disputed by legislators. When it came to health care reform, for example, "the president's own objectives were murky," Ganz said, which may have made mobilizing his supporters more challenging.
It's also hard for citizens to feel like writing to their members of Congress actually makes a difference. "I think one of the general challenges is that individual citizens have a much less direct role in the legislative process," said Sasha Issenberg, author of The Victory Lab: The Secret Science of Winning Campaigns.
Swaying individual legislators is less effective these days than it has been in the past, given how few moderates are left and how few members are willing to buck the party leadership, said John Hudak of the Brookings Institution.
To "go public" and try to build support for a given policy is seen by many political scientists as a president's last resort, Hudak added. If a president has to go to the people, it's usually because he's not getting much traction on Capitol Hill.
A bully-pulpit strategy would be most effective once legislators reach a bipartisan framework -- not while they're working toward one, Ornstein said. If a plan emerges, Obama should "immediately endorse it, bring everybody to the White House, praise it, praise them, and then go to the public," Ornstein added, noting that building public support could help give Boehner the cover he needs to call a vote in the House.