It is not uncommon for iPhone to come up with Apple ID or Apple Account Verification pop-up after installation of an update and at other times. In most cases, the pop-up goes away and does not return back when you follow the instructions to enter the password associated with your Apple ID.
If the pop-up keeps reappearing, even though you have entered the correct password, the reason could be due to a temporary glitch, outdated payment information or due to security reasons.
Repeated “Apple Account Verification” Pop-ups on iPhone
In most cases, the issue of an iPhone coming up with repeated “Apple Account Verification” pop-ups can be fixed by updating payment information, signing out of iPhone/signing back in and other methods as provided below in this guide.
If you are unable to fix the problem using these steps, it is possible that your Apple account has been placed on hold by Apple and this can only be resolved by contacting Apple support.
1. Enter Correct Password
It is possible that you are not entering the correct password associated with your Apple ID.
1. When Apple ID or Apple Account Verification pop-up appears, tap on Settings.

2. On the next screen, make sure that you are entering the correct password and tap on Sign In.
2. Check Your Payment Information
Make sure that the Credit Card or other form of payment information linked to your Apple ID correct.
1. Go to Settings > tap on your Apple ID Name.

2. On Apple ID screen, tap on Payment & Shipping > If prompted, enter your Apple ID Password.
3. On the next screen, tap on your Credit Card or the listed Payment Method and make sure it is up to date.
3. Sign Out And Sign-in to iPhone
This step can help, if the problem is being caused due to an iCloud glitch or conflicting Apple IDs.
1. Go to Settings > tap on your Apple ID Name.
2. On the next screen, make sure that your Apple ID (Email Address) is correct, scroll down to bottom of the screen and tap on the Sign Out option.

3. Follow the instructions to Sign-out from iPhone.
After you are completely signed-out, wait for 30 seconds and restart your device by pressing the side or power button.
4. Install Updates
Make sure that your iPhone is up-to-date with the latest iOS update.
1. Go to Settings > General > Software Update.

2. Wait for iPhone to search for updates > tap on Install Updates (if an update is available).
5. Force Restart iPhone
Force Restarting the device has been listed as a potential fix for this issue on Apple support forums and this certainly can help in clearing out temporary bugs which might be causing this issue.
Force restarting of iPhone can be achieved by pressing and releasing Volume up, Volume down and Side or Power button in a sequence, as provided in this guide: How to Hard Reset iPhone (All Models).
6. Login to Apple ID Page
If the problem is not yet fixed, login to your Apple Account on a MacBook or Windows computer and see if there is a need to Reset Apple ID Password or update/re-confirm answers to security questions.
1. Using a Mac or PC go to account.apple.com.
2. Enter your Apple ID and Password to login to your Apple Account.
Once, you are logged-in, you may see a message requesting you to update your Apple ID password or confirm your answers to Security questions.
3. Follow the prompts to Change/Update your Apple ID password or Reset Security Questions.
After you are done with this, you may still find iPhone coming up with “Apple Account Verification” pop-up, the first time you go back to your device and also while trying to use iMessage.
This happens, due to your Apple ID being linked to all iCloud related services. Once, your Apple ID is verified on two or more iCloud services, the problem should go away.
7. Reset All Settings
If above steps did not help, you can Reset all the Settings on your iPhone to their original or default settings and see if this helps in fixing the issue.
Note: This step does not remove any data from your device.
1. Go to Settings > General > Transfer or Reset iPhone > select the Reset option.
2. On the pop-up, select Reset All Settings option.

3. Enter your lock screen Password > On the confirmation pop-up, select Reset All Settings option to confirm.
Wait for all the Settings on your device to be reset to their original or default settings.