The requirement is for private communication between a Cloud Run service and a GKE API, following best practices.

Option A exposes the GKE API to the public internet, which violates the "privately reach" requirement. Relying on dynamic IP allowlisting with Cloud Armor is complex and less secure than private networking.

Options B and C configure overly permissive firewall rules (allowing all egress or ingress) and do not establish the necessary private network path between Cloud Run (which normally runs outside your VPC) and the GKE cluster within your VPC.

Option D describes the standard Google-recommended pattern for this scenario:

Internal Application Load Balancer (ILB): Expose the GKE service (API) using an ILB. This gives the service a private IP address accessible only within the VPC network (or connected networks).

Cloud DNS: Create a private DNS zone and record pointing a fully qualified domain name (FQDN) to the ILB's private IP address. This allows services to reach the API via a stable name instead of an IP.

Serverless VPC Access Connector: This connector creates a bridge allowing serverless services like Cloud Run to send traffic into your VPC network.

Cloud Run Configuration: Configure the Cloud Run service to use the VPC Access connector. The application code can then call the GKE API using its private FQDN registered in Cloud DNS.

This setup ensures traffic flows entirely over private networks (within the VPC via the ILB and through the VPC Access connector), meeting the private communication requirement securely and reliably.