First, try blocking notifications for the suspect app.
Pull down your notification screen, and long press the pestilence.
Then gently, slowly, move your touching finger to the left or right.
A pinwheel should appear, that will take you directly to the notification settings of the offending app.
If this fails, then decide if this app is really anything you need or want (regardless if it is a system app).
You should be able to do something, such as 'uninstall all updates', and/or eliminate permissions.
Personally, I have seen system apps that eventually received updates that put ads into apps that did not have ads before.
So your experience, as described, is not unheard of.
If uninstalling all updates and denying permissions DOES work, then you must be vigilant to not allow the app to ever be updated again, or the problem will begin again.
This means that if you use Google Play Store, you must have the 'auto-update' setting OFF.
The same for any other app store you may be using.
If you uninstall all updates and the problem is NOT solved, then two things...
1. it is probably not that app doing it
2. you need to find out what app is doing it
Do the notifications pop up when you are using the device?
If so, immediately check
Settings
Developers options
Running apps
and see what is running.
Another method is to use an app such as No Root Firewall, which has a log that tells you what apps connect to the internet and when.
If No Root is running, and you get a notification, check the log.
Then you can use the Apps section of No Root to block internet access for the app.
You may find that the features provided with No Root are useful with other apps, as well.
It is a very useful tool.
Try these things first and then get back with us with the results.