Changing Sub Domain name servers

Hello All,

We own a domain, for example

They have current A record that point to a IP.

My client has requested to change the name servers of (example) to different name servers but not to change the name servers for, when I try add into cloud flare it doesn’t want to add the site.

From my understanding there are no name servers for they can only be A or CNAME records etc.

Is anyone able to give me incite on how I can set the name servers for


You can add NS record(s) for ‘www’ that point to those other name servers.