When you login to wp-admin of TurnKey WordPress VM, you will see updates for WordPress, Plug-ins, and Themes in [Dashboard] → [Updates]. When you are ready for installation, it will prompt for ftp user name and password. It never works with your root account.
It’s not because of ftp connection type, you may stay with FTP or FTPS (SSL). It’s due to the access right of www directory. More about access right may be found in Secure wp-config.php section in another post.
Please login to the shell or use Web shell as root and type following command to change the owner and group of /var/www and its sub-directories to www-data
chown -R www-data:www-data /var/www
Now, you may upgrade within wp-admin successfully!