Yellen remains unlikely to get the nod for a second term for several reasons but her renomination is not impossible. The case for Trump renominating Yellen is really not that hard to make: She's as dovish a candidate as self-proclaimed "low rates guy" Trump has on his list of finalists. The markets would love status quo at the Fed and Trump loves his market rally.
Does he really want the possible sell-off that could follow picking a hawkish candidate like Stanford's John Taylor?
If he renominated Yellen, Trump would also win a bunch of love from Senate Democrats, who he may need on tax reform, while his base probably wouldn't care that much. After all, while some of his hardcore nationalist base cares about the Fed and distrusts it as a "globalist" institution, it is far from their animating issue.
Trump would probably get a day or two of bad headlines in Breitbart if he renominated Yellen but that's probably about it. Senate Republicans, especially on the banking committee, would hate a Yellen renomination. But Trump does not really care about how Republicans react to his decisions.
It's also very difficult, if not impossible, to imagine Senate Republicans rejecting Yellen for a second term. Doing so would infuriate the president and rock markets. If Yellen gets renominated she's likely to get confirmed quite easily.
Still, as one administration official told me on Tuesday, it remains highly unlikely that Trump sticks with the current Fed chair for one main reason: Former President Barack Obama, who nominated Yellen as the first female Fed chair in 2013.
If anything animates Trump's decision-making it's reversing anything Obama has done — from health care to immigration to economic policy. Yellen is likely to benefit from being the last candidate Trump speaks to about the Fed chair job. But it seems unlikely to many close to the process that she can win over Trump enough to overcome his instinct to dismantle the Obama legacy in every area he possibly can.
So put the odds of a Yellen renomination at somewhere around 25 percent, if not lower. That leaves Taylor, Fed Governor Jay Powell (a favorite of Treasury Secretary Steven Mnuchin), former Fed Governor Kevin Warsh and National Economic Council Director Gary Cohn on the short list. If there is a dark horse candidate, nobody knows who it is.
Both Taylor and Warsh had strong interviews with Trump, but sources say both Mnuchin and Trump are concerned about Warsh's age. At 47, he'd be one of the younger Fed chairs in history and Trump typically likes august, silver-haired types. That's partly why Taylor is rising in the betting line and Warsh is receding somewhat.
Speculation that Trump could go with Warsh as chair and Taylor or Powell as vice chair persist. It's not clear, however, that either would settle for the No. 2 slot — Warsh definitely would not.
No decision on the Fed chair job is expected to come down this week. Trump still needs to have his meeting with Yellen then sit down with senior staff and Mnuchin to make the final call. That could happen next week or slip toward the end of October or early November. But probably not later than that. The White House wants to get the decision out of the way and ensure that a hearing and confirmation vote can take place before Yellen's term ends in February.
It's really impossible to call right now who it will be. The current pecking order is probably Taylor and Powell tied for first, followed by Warsh, Yellen and Cohn. Some White House and Hill sources say Cohn is completely out of it after his break with Trump on the Charlottesville, Virginia, protests. Others say don't write him off just yet.
The only way to know for sure right now would be to climb inside Trump's head. And that wouldn't really work either because, at this point, even he doesn't know.
—Ben White is Politico's chief economic correspondent and a CNBC contributor. He also authors the daily tip sheet Politico Morning Money [politico.com/morningmoney]. Follow him on Twitter @morningmoneyben.