Follow the instructions and press that number. You will be asked to delete your current Voicemail greeting. Click on the ‘Yes’ button that appears on the screen if you want to remove that. You can save your new recorded Voicemail greeting or can use the default recorded option. From the home screen, select the Phone app. Note: Alternately, you can set up voicemail by selecting the Phone app then selecting and holding the 1 key. Select Keypad, then select the Visual Voicemail icon. Select Start. Select Ok. From any Home screen, tap the Menu key. Tap Settings. Tap Call settings. Tap Voicemail. Tap Voicemail again. View the voicemail number. Press and hold the 1 on your device’s dial pad to call and connect to your voicemail. Press the star * key to access the main menu. Press 4 to access the mailbox options. Press 7 to change the language of the voicemail prompts to either Spanish or English.
1. Press the fixed key with the envelope graphic, or dial the phone’s phone number.

Still, if you are unable to record a custom voicemail greeting on iPhone, then its time where the carrier comes in the picture then for their assistance and get it fixed, the custom voicemail not recording issue.
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.