At the moment Safari on iOS doesn’t support web push notifications.
However Apple has just announced that Safari 11.1 supports Service Workers. Service Workers are a fundamental technology in order to create web push notifications, so this is a step forward.
2022 Update: Apple is working to implement web push notifications on Safari and iOS and it is currently possible to enable the Push API as an experimental feature. Check out this post for more up-to-date information.
In order to make web push notifications work in Safari on iOS Apple have to implement Push API as well, which depends on Service Workers tech.
Therefore, we are not so close as we think 🙂
https://caniuse.com/#search=push%20api