Can we use or integrate WordPress with ASP.NET?

You could run WordPress using Phalanger, which can execute PHP code on the .NET platform (faster that standard PHP!) and allows mixing the PHP and standard .NET languages/frameworks in a single application.

PeachPie is an open source PHP compiler to .NET. This supports ASP.NET and ASP.NET Core, and it is performant.

Yes, it is actually possible.

Check out WP.NET:

WP.NET runs WordPress compiled on the robust, fast and secure .NET Core framework using the cutting edge PeachPie compiler platform.

WordPress is written in PHP and runs on Apache, so you can't really run it in ASP.NET. There are methods however that enable you to run it on IIS.

The best way to go with integrating WordPress and ASP.NET is by using one of the supported XML-RPC protocols. You can see more about it here:

XML-RPC Support