Keeping track of client logged-in OS user when connecting to a database server

We are implementing a monitoring solution that keeps track of logins to SQL Server. We need to capture the following event: a user logs in to a Windows client and from there logins to a remote SQL Server using SQL Authentication. When this occurs, we need to keep track of the Window user that logged into the Windows client. Is there any chance to configure SSMS to pass the credentials of currently logged-in Windows user to the SQL Server it is connecting to?