I’m not seeing Trainer Battle invites or the Battle icon on the Friend screen. – Pokémon GO Guide

If you are not receiving Trainer Battle invites or you cannot see a battle icon on your friend screen, check the following settings:

Visit the Settings page and enable "Allow Battle Challenges with Friends". If this is unchecked, your friends will see a "Player is currently not receiving challenges" message when they try to challenge you.

Visit your device’s date and time settings and set your device clock to sync automatically with local time.

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Pokémon GO Game GUIDE