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

Checkout this article for full details.

Add the following to wp-config.php:

define( 'FS_METHOD', 'direct' );

Let me know how it works for you.

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.