The question is what exactly would that be?
The fact is that Hagel has already endured plenty of hits against his nomination, including his rocky past with his political party, revelations about past statements that were portrayed by critics as anti-gay and anti-Israel, and what was widely regarded as a very poor performance in his confirmation hearing.
Given what he has been through already, both sides agree that for Hagel's confirmation to be thrown into doubt it would take something really big.
"Republicans are smart enough to realize that they can't overplay their hand here; after all we are talking about the next secretary of defense," said Jim Manley, a former top aide to Senate Majority Leader Harry Reid (D-Nev.). "They accomplished what they set out to do, which was to bloody up another Obama nominee, so unless a real bombshell comes out in the next few days, I expect that there will be votes next week to confirm Senator Hagel."
Added a top Senate GOP aide: "It's highly likely he'll be confirmed next week." That aide and others interviewed for this story asked to be anonymous to discuss strategy.
With that in mind, here are the few things that could change that:
* A big revelation about Israel: Republicans continue to unearth Hagel's past statements about Israel in hopes that they will inflame pro-Israel groups enough to come out against Hagel -- which would suddenly put some pro-Israel Democrats in a tough spot. But that's proven difficult. Republicans late last week pointed to a story in the Washington Free Beacon about a Rutgers University professor who said Hagel in 2007 called the U.S. State Department an adjunct of the Israeli foreign minister's office.
The problem is that it's based on the recollection and notes of one professor, there is no direct quote, transcript or recording of the event, and now another professor who was present says Hagel said no such thing. The story so far has gotten little traction.
If anything takes Hagel down, though, it would likely be something along these lines -- a revelation that is just too hard for pro-Israel groups to give Hagel a pass on.
"If yet another comment or speech comes out tomorrow, then you might see folks getting nervous," said one anti-Hagel Senate GOP aide. "And then another question is whether that forces AIPAC to end its silence, which would be a game changer. On the other hand, if the oppo well is dry and that's all there is, he probably is okay."
* Ethics: The one thing that has not stung Hagel is any hint of corruption or wrongdoing -- a big reason his nomination is still on track.
Policy differences are generally not seen as a good enough reason to deny someone confirmation as a Cabinet secretary or a Supreme Court justice -- and that goes double for a defense secretary nominee. In order to really make Hagel unpalatable, Republicans would need something that casts doubt on his character rather than his political positions. As of now, Hagel is a war hero who gets the benefit of the doubt when it comes to that stuff.
We need to stress here that there really is no reason to believe there is anything ethical that could drag Hagel down, but it is certainly one of the few things that would change the game. And it's territory the GOP hasn't really penetrated yet.
* A big-name Democrat comes out against Hagel: Democrats have been united behind Hagel -- even as several of them initially expressed reservations about his nomination. Once Sen. Charles Schumer (D-N.Y.) said he would support Hagel's nomination, that pretty much foreclosed the possibility that the anti-Hagel contingent would be bipartisan.
This has prevented Republicans from arguing that opposition to Hagel is anything but a partisan political play. And their case is hurt by the fact that even some Republicans -- Sens. Mike Johanns (R-Neb.) and Thad Cochran (R-Miss.), in particular -- are now supporting Hagel.
It's very difficult politically to hold up a defense secretary nominee, and that goes double when no Democrats are on board and even some Republicans are supporting the nominee. If for some reason a Democrat got cold feet -- however unlikely that is -- it would surely embolden Republicans to keep up the fight.