AT&T: If you’re an AT&T customer, you have the option of creating your own basic voicemail. You’ll need to press and hold the 1 key down while you’re on your phone. AT&T will then ask you to set up your greeting via voice prompts.
Open your phone’s voicemail app, then tap (or in some cases, tap and hold) the message you want to save. You should be presented with a list of options; the save option will usually be listed as “save”, “save to phone,” “archive,” or something similar.

Another worth considering trick is to remove all the old voicemails that are consuming the storage and creating multiple dilemmas like this. To delete the voicemail here are the steps, Open the Phone app. Tap on Voicemail at the bottom tab. Then tap on Edit at the top-right screen. Select All the Voicemails and at last tap Delete.
To change the voicemail password of your iPhone, follow the following easy steps. Step 1. On your iPhone, go to settings and click "Phone". Step 2. On the Dropbox that will appear, click "Change Voicemail Password". Step 3. When prompted, enter your current password and click "Done". Step 4.