Apparently, it's something that somehow got into your website.
Things to try then:
1. Back to the latest backup when it wasn't yet happening,
2. FTP to your site and see if you can find the cause in one of the files there
3. Ask tengointernet for help
I help manage a WordPress site, and recently we've been having issues with a PopUp from TengoInternet. Here's a screenshot of the pop-up.
It basically asks us for a username and password, saying "Authentication Required"), for dashboard.tengointernet.com.
It'll pop up for some folks every time they go to our website or try to navigate within our site (FYI, its www.hartsmill.org). If we hit cancel the popup will go away, only to come back again when another of our site's pages is viewed. We've tried clear caches and cookies, but the problem still persists. We're not sure why the problem started - apparently TengoInternet is a service provider for RV parks?
If anyone has any knowledge on why this is happening, or how we can prevent it from popping up, I'd really appreciate it!