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.
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