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