See here. Here’s the upshot:

Given these memos, though, along with President Obama’s strong words on Citizens United (citing it as a central reason he tapped Kagan), it’s fair to assume that Kagan will approach campaign finance issues from the perspective that most regulations are constitutional.
It also seems fair to assume that President Obama would not be making Citizens United a central element of Kagan’s nomination if he hadn’t received some assurances that she generally shares his views on campaign finance regulation.

For reasons I’ve expressed (see here, here, here, and here), I disagree that we really know anything about how SG Kagan would vote in a case like Citizens United. It would not surprise me if she voted like Justice Breyer; it would not surprise me if she voted like Justice Scalia.
I further would be shocked if the President had received “some assurances” on her views on campaign finance. I just don’t think presidents explicitly ask for such assurances, if for no other reason than that someone will eventually ask the nominee if she has given such assurances to anyone about such cases, or communicated her views to anyone about issues likely to come before the Court. I’d add that I’m not sure even where President Obama stands on these issues, despite his recent anti-CU comments, which I believe have a certain electoral advantage. We know that the President’s chief counsel, Bob Bauer, has in the past advocated the deregulatory approach of the CU majority. (Remember as well that despite the campaign promises, we’ve yet to see the President come out with a fix for the public financing system for presidential campaigns.)