Voicemails are recorded on your network as audio files and are downloaded to your phone. They are then played back in the usual way, just through the phone app rather than through the music app. Any voicemails you have are visible in the Voicemail app and should be able to be played back by pressing the play icon. This doesn’t always work, though. Here are some fixes you might try.
There is a way to save a voice message on iOS mobile by recording it on Notes or Voice Memos and set them as voicemail greeting. Also, you can share them using SMS, Email, Airdrop, Google Drive, and more, to make a back-up of the recorded voicemail. It isn't delightful when you expect a client call and can't answer it at the moment. You can't get all your phone calls at all times, especially when you are busy with your work. That is why it is important to record voicemails on your iOS device and set it as a response on an incoming call. Therefore, we have listed the steps below on how to capture voicemails on iOS.

Step 4. Begin recording your custom voicemail message by tapping “Record.” You can also listen to your recording by tapping “stop” and “play” before proceeding to the next step.
At present, you have to record your greeting by recording directly into the iPhone microphone -- the results of which are average, of course.