to perform the requested action wordpress needs to access your web server. please enter your ftp

This means that WordPress is having limited permission for making changes in the folder that it was installed.

In-order to fix this, all that you need to do is provide necessary permissions for the same.

Run the following Command in your Terminal / Putty / Commandline Prompt after connecting to your Server via SSH:

sudo chown -R apache:apache /var/www/html

In my case, I solved this by switching from GIT back to FTP mode.

No more warning.

Perhaps that'll help somebody else too.

Although the question is not that new anymore I want to add my two cents on this issue also.

A lot of ppl have Centos(7) on their VPS server and following code lines could solve their problem.

Imho has all to do with SELinux which withholds WordPress from doing it's job as wished. It goes to far to explain what SELinux is and what it does. FYI the introduction starts with:

Security-Enhanced Linux (SELinux) is a mandatory access control (MAC) security mechanism implemented in the kernel.

Only 3 steps to folow:

• 1 Open a terminal (or access the server through SSH)
• 2 Add following code line chcon -R -t httpd_sys_content_t /var/www/html/wordpress
• 3 Add second code line chcon -R -t httpd_sys_rw_content_t /var/www/html/wordpress

No reboot from the server or restart from any daemon needed.

I won't say it helps everybody but for those who didn't disable SELinux it should be a relieve.

Cheers

Note: Please adjust to your own needs (meaning path to WordPress)

edit: be sure to remove the line define("FS_METHOD", "direct"); when it is/was used in wp-config.php because that's absolutely a no go when above code lines do as wanted.

Even though it is totally correct to have the ownership as root:apache with permissions 775, and httpd to run as apache, Wordpress does not like this. It wants the owner to be apache, as per wp-admin/includes/file.php:

    // Attempt to determine the file owner of the WordPress files, and that of newly created files
   $wp_file_owner = $temp_file_owner = false;
   if ( function_exists('fileowner') ) {
      $wp_file_owner = @fileowner( __FILE__ );
      $temp_file_owner = @fileowner( $temp_file_name );
  }

Yours would be:
wp_file_owner = root
temp_file_owner = apache

if ( $wp_file_owner !== false && $wp_file_owner === $temp_file_owner ) {
    // WordPress is creating files as the same owner as the WordPress files,
    // this means it's safe to modify & create new files via PHP.
    $method = 'direct';
    $GLOBALS['_wp_filesystem_direct_method'] = 'file_owner';
} elseif ( $allow_relaxed_file_ownership ) {
    // The $context directory is writable, and $allow_relaxed_file_ownership is set, this means we can modify files
    // safely in this directory. This mode doesn't create new files, only alter existing ones.
    $method = 'direct';
    $GLOBALS['_wp_filesystem_direct_method'] = 'relaxed_ownership';
}

If $wp_file_owner is same as $temp_file_owner then proceed. Yours would be caught in the elseif, which according to the comment does not allow delete/create, but only updates (I verified this by updating the code of a plugin from within Wordpress, and it worked).

Note I did not extensively look through the code, this is just my quick interpretation. I had the same problem and once I switched user:group so that the httpd user is also the file owner, it did not prompt for FTP credentials anymore.

Add the following to wp-config.php:

define( 'FS_METHOD', 'direct' );

Let me know how it works for you.

Not a direct answer, but probably has to be said - this is one problem you should avoid solving unless you are talking about a local development in which case you can just set permissions to 777.

The reason is that if the webserver can overwrite your code, then any malicious code running on it will be able to do that as well. The risk is just so much bigger than the convenience of saving few seconds by not having to type the ftp credentials.