___________________
Method #1. Perhaps the easiest way to check is to call the number *639# (*NEW#) directly from your iPhone. Shortly, you'll receive a text message from AT&T. Here's how it looked on my iPhone 5s.
Method #2. Another way to check is to use Apple's Apple Store app. On the home (Featured) page is an entry called “Upgrade your iPhone.” That takes you to a page where you can also check your eligibility for a new phone. One wrinkle is that, for security, you'll be asked for the last four digits of your SSN. If you'd rather not provide that in an app, Methods #1 and #3 don't require it.
Use Apple's “Apple Store” app.
Method #3. From your Mac, PC or tablet, you can log on to your AT&T wireless account via a browser. Recall that your User ID is your mobile number. (Personal information has been scrubbed from the screen shots below, so your pages will look slightly different.)
After you're logged in, from the Overview page (tabs at top), in the drop down list labelled “I want to…
From the “Overview” tab…
… select “Manage my plan & services.” Then click “Check upgrade eligibility.” You'll see the something like this depending on how many iPhones you have on your account. Click on “Upgrade Eligibility” for the desired phone.
Eligibility for each phone on the account.
Note that if your iPhone is eligible for an AT&T upgrade, you have two choices now. The first is the traditional two-year contract extension. The other is AT&T's new “Next” plan that allows you to get a new iPhone/smartphone every 12 months by paying for the phone itself with a pro-rated fee each month.
Finally, we're checking to see if anything is different when checking upgrade eligibility with Verizon. For quick reference, here's the Verizon article from last year.