Is it possible to set 'Forwarding Numbers' through API?

Not via the API, but this can be done from within the Online Account Portal (https://service[.devTest].ringcentral.com).

Thanks for the quick reply.

Can you explain a little more about what that means? Does that require manual input?

There is an advanced-level API to manage forwarding numbers (which you may not be able to see in API docs). But you can try (it requires EditExtension app permission):

POST /restapi/v1.0/account/255537016/extension/255537016/forwarding-number HTTP/1.1
Content-Type: application/json
{
 "phoneNumber": "+16504326573",
 "label": "Custom number"
 } 

There are also APIs to modify and delete forwarding numbers (usual PUT and DELETE). You can send a support request to get access to full documentation about forwarding numbers API (and other advanced APIs) for your account.