Whisper
Whisper is a call-handling feature that lets a supervisor speak privately to the agent on a live call without the caller hearing. The same word covers a short announcement played to the answering party before a call is connected.
Also called: call whisper, whisper coaching, agent whisper
The word covers two related things. In supervision, whisper sits between monitor (listen only) and barge (join openly): the supervisor can coach the agent, "offer the discount", while the caller hears nothing.
In routing, a whisper message is the brief announcement played to whoever answers a forwarded call before the two sides are joined: "Call from the website, plumbing inquiry." It gives the recipient a second of context and lets them accept or reject the call.
Both uses are about giving one side of the call information the other side does not hear.