One option would be to use a WebSocket server running parallel to your Django server which has a REST/Push API, and then do pushes from Django by simply HTTP/POSTing to the WebSocket server, which in turn delivers the messages to all connected WebSocket clients.
That way, you don't need any structural/technical changes to your existing Django app, nevertheless have a scalable, modern WebSocket based push feature.
For a hosted service providing above, I'd have a look at http://pusher.com.
If you prefer running your own, I'd have a look at http://autobahn.ws, which is deployed as a ready-to-run virtual appliance (VMware, VirtualBox, Amazon EC2). You can find working examples of REST/API here https://github.com/tavendo/AutobahnPushPython
Disclaimer: I am author of Autobahn Open-source, and work for Tavendo, which offers Autobahn.ws (the commercial offering based on Autobahn OSS).