This is one settings that is really missing in Drupal (both 7 and 8).
It was really annoying because (almost) every admin page has this Update notification message if you don't have the latest modules and you want to keep Update module enabled because of Drush up.
The goal of this issue is to hide this notifications in Update module's settings page - /admin/reports/updates/settings
For all users.
Like this.
Of course it will be enabled by default but at least you can disable it to stop spamming you while developing.
P.S. The similar patch would apply to Drupal 8 too.
EDIT:
Drupal 7 patch in #1
Drupal 8 patches from #2 on.
Alternative approaches
There are three different approaches to this problem, to be discussed which one to use finally. Afterwards both others should be closed as duplicate:
#3239970: Show update notifications only with permission "administer software updates"
#332796: Add permissions to the update.module to hide warnings
#2059375: Update notification messages as an option in settings (THIS)