401 Unauthorized error web api mvc windows authentication

I suspect that the two web services may be hosted on the same server. In this case, the problem may be caused by the loopback check. In order to test, try referencing the service without using the fully qualified domain name and see if it works. If it does, use the following steps to specify which host names are on the local computer.

Method 1: Specify host names (Preferred method if NTLM authentication is desired) (https://support.microsoft.com/en-us/help/926642/)

To specify the host names that are mapped to the loopback address and can connect to Web sites on your computer, follow these steps:

1. Set the DisableStrictNameChecking registry entry to 1. For more information about how to do this, click the following article number to view the article in the Microsoft Knowledge Base: 281308 Connecting to SMB share on a Windows 2000-based computer or a Windows Server 2003-based computer may not work with an alias name
2. Click Start, click Run, type regedit, and then click OK.
3. In Registry Editor, locate and then click the following registry key: HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\Control\Lsa\MSV1_0
4. Right-click MSV1_0, point to New, and then click Multi-String Value.
5. Type BackConnectionHostNames, and then press ENTER.
6. Right-click BackConnectionHostNames, and then click Modify.
7. In the Value data box, type the host name or the host names for the sites that are on the local computer, and then click OK.
8. Quit Registry Editor, and then restart the IISAdmin service.

https://docs.microsoft.com/en-gb/archive/blogs/sharepoint_foxhole/disableloopbackcheck-lets-do-it-the-right-way

**Edited to be in the form of an answer and include detailed steps from referenced links

If you are using WebClient you need to set the Credientials. How are you calling the web api from the windows service?