I have a solution with contains a class library targeting the .Net framework 4.7 and another project with ASP.NET Core 2.2 .
The class library targeting the .Net framework contains controllers as shown below with the route prefix and the route name
And then in my StartUp.cs i have configured the below

On running this i make a call to the api in the .Net framework class library the call does not return the response from the web api instead returns the default index.html content.


I see on the first screen classes like
RoutePrefix,ApiController,HttpResponseMessage. They are from classicAsp.Net WebApi. It is impossible to use old ApiControllers inAsp.Net Core. It means you need to rewrite you controllers usingAsp.net Core.Asp.net core projects may targeting to both
.net coreand classic.Net Framework. If you really need to target to the.Net Frameworkyou can create a class library targeting the .Net framework 4.7, create thereAsp.Net Corecontrollers and then use that library inAsp.Net Coreproject as an external dll.