I recently updated the PHP version on my Amazon Linux AMI and then a few days later notice I could not download WordPress updates. I was getting the following error:

Download failed. cURL error 77

If you get this error you might try restarting your web server process and / or php-fpm if you are using it

So for me the commands I needed worked out to

service nginx restart
service php-fpm restart

This was the content of the log when the cURL error was being thrown (It normally appears on one long line in the long but has been wrapped for readability)

2018/10/30 23:24:56 [error] 3246#0: *130467 FastCGI sent in
stderr: "PHP message: PHP Warning: An unexpected error occurred.
Something may be wrong with WordPress.org or this server’s
configuration. If you continue to have problems, please try the
support forums. (WordPress could not establish a secure
connection to WordPress.org. Please contact your server
administrator.) in /var/path/to/webroot/web/blog/wp-
includes/update.php on line 347