1) President Obama really likes Summers. And he's surrounded by Summers's longtime colleagues and friends. Conversely, Obama doesn't really know Yellen, and nor do any of the White House's economic principals.
2) The Obama administration's top concern is choosing someone who cares about the Federal Reserve's mandate to maintain full employment as well as its mandate to keep inflation low. Yellen and Summers are both seen as clearing that bar. So the choice is defaulting to other considerations.
3) This White House, more so than any other in modern memory, knows in its bones that the economy can fall apart at any second. China could suffer a hard landing, Europe could fall apart again. Some London Whale-like trader could blow a hole in JP Morgan Chase. If that happens -- particularly given Washington's dysfunction -- the Fed is really the first responder. This White House is very comfortable with how Summers handles a crisis.
4) There's also a feeling that the chair of the Federal Reserve can do more if he or she is truly trusted by markets. Rightly or wrongly, there's a sense that Summers has the market's trust in a way Yellen doesn't.
5) The big open question is Summers's ability to manage the Federal Reserve's Open Markets Committee. Here, Summers's reputation for being difficult to work with is a big issue. But inside the White House, that reputation is considered overblown, or at least outdated -- after all, they worked with him, and enjoyed working with him, and there's some sense that maybe a more aggressive Fed chair wouldn't be the worst thing in the world.
That's not to say Summers is anywhere near a sure thing. His confirmation would be far tougher than Yellen's, as Republicans will make him answer for the stimulus and the bailouts, and progressive Democrats have a list of grievances going back to financial deregulation in the Clinton-era. There's also the simple fact that appointing Yellen would break a significant glass ceiling -- and do so in an administration that hasn't always been great about appointing women to top economic positions. And Summers continues to be a polarizing figure: Those who like him love him, but those who don't like him really don't like him.
Against all that, the conventional wisdom -- which I fully bought into -- a month or two ago was that Summers had little real chance. The politics of it just didn't make sense. But if Obama feels strongly about Summers and his qualifications, the Fed job is more than important enough for him to fight through the politics. And so, though I'm a bit surprised to be saying this, at this point my reporting says Summers is the front-runner.
For more on this, here's Paul Krugman's endorsement of Yellen, and Tyler Cowen's endorsement of Summers. I'll just say, for the record, that I really don't know who should run the Federal Reserve, and this post isn't an endorsement of either candidate.