It looks like it might be possible with Aria, but you're going to need to read the spec yourself and do testing--how Aria is presented isn't really specified and the implementation status of Aria varies wildly across browsers and screen readers.
It is easier and much more sensible to just provide visible text below it in DOM order, for example.  Otherwise, you'll need to do some testing and experimentation-which will be more time-consuming than just providing some text below it.
Nothing has hurt accessibility quite as much as using divs for everything.  This is probably why we have Aria.

Alt attributes generally only work on images, sadly. Assigning ARIA roles based on JavaScript might work, or you could append some text DOM nodes that are only visible to screenreaders to determine if there is damage or not.

HTH,
Balliol

You can "just" add aria-label.  But you missed my entire point.  That's not necessarily implemented everywhere nor guaranteed to be presented in the way you think it is.  The standard gives almost no guarantees on that regard.  Have you actually tested this with screen readers and, if so, which?
The thing people don't understand about Aria is that the AT market moves slowly.  Every time I bring it up, someone jumps on it like it's the end-all solution for everything.  If you're doing proper testing across multiple browsers and screen readers, go ahead.  If you're not doing proper testing, please just stop now and think about all the harm that making everything in your entire game a div does in terms of accessibility.  At worst, your new found solution of Aria will actually break things, especially if you ever go beyond Aria-label--I have seen this many times, most notably with Google.

It might be. It might not.  It's going to depend on the screen reader and the platform and whether you ask me that question now or in 6 months.  I read the algorithm for the computation of text that Aria's spec provides and it's not exactly simple. Or short, for that matter.  It almost certainly works with NVDA and Firefox, and I'd be willing to bet quite a lot that it doesn't work well or at all with Voiceover on OS X.  To get some idea, consider all the HTML5 features and how long it's taking for them to all be consistently implemented everywhere.  Now cut the user base down to like 1% and factor in the fact that Jaws is money-driven, hasn't done anything major in years, has their hooks in with governmental agencies, and is still the most popular screen reader.
I'm sorry if I'm snappish here.  Sighted devs find Aria and then assume that it's the end-all and they can go right ahead with whatever practice is causing the problem in the first place, which is why we have things like Angular.js that provide no end of trouble without special extensions.  The problem with Aria is that it is almost exclusively for accessibility.  As a consequence, it's actually really fuzzy where the browser's responsibility ends and the screen reader's responsibility begins; the implementation involves both.  Basically, it's assumed that if you're using Aria you know what you're doing, so the end user can't usually override this.  In fact, overriding application role took a very large discussion on the NVDA tracker, and is still a major problem.  Aria-label is relatively safe, in that the way it's likely to fail is simply that the screen reader doesn't respect it, but going beyond that will almost certainly cause you to start running into bugs that may render something marginally usable unusable, or "fix" something that worked fine in the first place into a state where it doesn't anymore.
The real fix to these problems is to get people back onto the content and presentation separation idea and off the CSS and Divs for everything; if I could figure out how to do this, I would be very happy.  When Aria comes up, however, sighted devs tend to use it as an excuse to go right ahead without any actual testing with the technology that Aria is supposed to make you work with, like Aria is some sort of magic super sauce of fixing.

It also doesn't help that googling for half an hour did not find much information on how to add supplementary info for screen readers.