Unknown cipher in list: TLSv1 SSL routines:SSL23_GET_SERVER_HELLO:sslv3 alert handshake failure
Upgrading CURL to the latest version 7.59.0 will fix the issue while accessing Cpanel/WHM. However, the curl upgradation has to be done manually as the official Centos repo does not ship the latest versions.
service httpd restart
To continue using PHP 7.0/7.1 in Centos 6/7, curl library linked in php web module has to be upgraded to the latest.
yum install libcurl-devel openssl-devel php-devel
wget https://curl.haxx.se/download/curl-7.59.0.tar.gz; tar -zxf curl-7.59.0.tar.gz; cd curl-7.59.0;./configure --with-ssl; make; make install
If php7.1 is running on the server, then download the corresponding php source code
tar -zxf php-7.1.16.tar.gz; cd php-7.1.16/ext/curl/; phpize; ./configure; make; make install; service httpd restart
If php7.0 is running on the server, then download the corresponding php source code
tar -zxf php-7.0.29.tar.gz; cd php-7.0.29/ext/curl/; phpize; ./configure; make; make install; service httpd restart