I keep getting "Apple ID Verification" have given my id over and over and it keeps coming up.

I keep getting a notice "Apple ID Verification" Enter the password for my email in settings . Push not now or push settings. I have done both, entered my password, that I have written down and that I have used even yesterday. It opens up settings then goes back to same asking for verification. I had previously been given notices of storage almost full. I then went in and tried to delete old files, mail, games, etc. Do not know what to try next. Did everything I could think of. At a loss now. Please help.