On iOS 9 and above, you can export voicemail from your iPhone as a note attachment, or as a voice memo. But it is not possible to save them directly to your computer, or to export more than one voicemail at a time.
With iMazing, you can easily save your voicemail as an audio file, to your computer or to an external drive.
Anyone who wants to easily save or backup voicemail to their computer or to an external drive.
Download and install iMazing.
To export your voicemail to your computer:
Launch iMazing and connect your device.
In the left sidebar, select Phone > Voicemail. To view your most recent data, you may need to refresh your data.
If this is your first time connecting your device, iMazing will initiate a backup to pull your voicemail.
Select the contact(s) whose voicemail you want to export.
By default, iMazing will export all the voicemail from the selected contact(s). If you only want to export specific voicemail, select them now.
In the bottom toolbar, click Export. This will open a save panel dialog box.
In the dialog box, select where you want to export your files, then click Choose.
Use the CTRL/Command or SHIFT keys to select multiple contacts or voicemail.
Double-click your voicemail to preview it.
Visual Voicemail is an an optional feature which enables viewing voicemail directly on your device, without dialling a mailbox. Not all carriers support Visual Voicemail: if your voicemail does not appear on your iPhone, please check this Apple Support article.
Voicemail audio files are in .amr format. If you are a Windows user, you will need Apple's Quicktime to play .amr files on your computer.