.NET and Microsoft decided to merge the MVC and Web API functionality, after the release of ASP.NET. If you are looking for the implementation of Web API with ASP.NET, Here is the way to accomplish:
Migrating Existing Web API Controllers by Compatibility Shim
You will have to alter one code to call AddWebApiConventions() if you use HttpResponseMessage in Web API controllers. It will help to know the proper serialize that response by wiring up the HttpResponseMessageOutputFormatter.
}Depending upon your routing with Web API. You will need to register routes with Startup class.
In this way, you can migrate from your Web API code to ASP.NET Core.
This sample controller helps you to use your ApiController within ASP.NET Core with compatibility Shim.