Safari on iOS supports web push notifications starting from iOS v16.4 (released in 2023). Previous versions did not support web push notifications.
History
2023: iOS finally supports web push notifications using the standard Push API. It’s the last major browser to add support for this feature. However it comes with several limitations and restrictions that can make it difficult to collect subscribers. Limitations include the requirement that the web app must be installed (i.e. add to Home Screen) in order to display the permission prompt for notifications. Another requirement is that the prompt must be triggered by a user gesture (i.e. click on a button).
2022: 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.
2018: 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