Both Ninject.Web.WebApi and Ninject.Web.Mvc register:
this.Kernel.Bind<HttpContext>().ToMethod(ctx => HttpContext.Current).InTransientScope();
this.Kernel.Bind<HttpContextBase>().ToMethod(ctx => new HttpContextWrapper(HttpContext.Current)).InTransientScope();
And when you try to resolve HttpContextBase Ninject thows System.InvalidOperationException with message Error activating HttpContextBase\r\nMore than one matching bindings are available.