Gmail Sign in Screen in Mail App

Opened safari , took the cursor on top right hand side & clicked on gmail .

A window appears .

in the above screen shot of empty field type your gmail id , another window appears , uncheck stay signed in is always recommended for security purpose , when ever you sign in .

Is this an , IMAP account & you have configured your web mail with apple mail app ( vulture logo ) .

And , when you go to system preferences > internet accounts are these options checked .

And what have you installed in your system , is there any anti - virus , third party apps ? please reply , I have never seen or heard this issue and its a strange activity .

if you are using the stuff , please uninstall them .

After installing them you can rebuild the mails : Mail for Mac: Rebuild mailboxes

Or a second method is there : Take back of all mails in V4folder : Method for how to copy in V4 folder .

Click on go on top menu bar , hold option key click on > library > select mail folder > right click on mail there itself > click on copy mail > right click on extreme right hand side in column B , that is below persistence info.plist > click on paste item .

Entire mails are copied to V4 folder .

Go to the corrupted IMAP account , select it & click on minus sign it is deleted .

Configure new IMAP account , go to top menu bar ,file > import mail boxes > import data from > select apple mail > click on continue > select the back drop folder then choose new .

When new IMAP account is opened , a new folder with all mails will appear in the mail side bar .You can rename the folder .

It's happening to me as well.

Every time I open Mail I have to insert the password of all the Google accounts.

You are opening gmail through safari , a pop window appears showing your gmail id and password , an option is there remember the password , you must have checked the box .