A company is building an API-based application on AWS to meet its process workflow needs. The API will be accessed by clients located in the company's on-premises data centers. The company has established an AWS Direct Connect connection between its on-premises environment and AWS. A network engineer plans to deploy the API as a private REST API in Amazon API Gateway and aims to ensure that clients can access the API endpoint using private communication.

What solution can the network engineer implement to allow clients to invoke the API without requiring any additional infrastructure setup?