Hey there billbeth1,
We want to make sure we understand; do you use Apple Music with your iPhone and iMac, or are you syncing music directly from a computer? If using music on your computer and not Apple Music, this would explain the missing Sync Library option.
You can only sync your device with one Apple Music or Apple TV library at a time. If you see a message that your device is synced with another Apple Music or Apple TV library, your device was previously connected to another computer. If you click "Erase and Sync" in that message, all content of the selected type on your device will be erased and replaced with content from this computer.
If you try syncing from your computer using the steps below, does that help?
1. Open a Finder window and connect your device to your computer using a USB cable. Learn what to do if you don't see your device listed in Finder.
2. After your device appears in the sidebar of the Finder window, click the device to select it.
3. If prompted, trust your device. You might be asked to update to the latest version of iOS or iPadOS.
4. Select the type of content that you want to sync or remove from the horizontal list. To turn syncing on for a content type, select the checkbox next to “Sync [content] onto [user’s] iPhone.” More syncing options will now be available.
5. Select the checkbox next to each item that you want to sync.
6. Click the Apply button in the lower-right corner of the window. If syncing doesn't start automatically, click the Sync button. You may also be prompted to authorize your computer to complete the sync.
Use the Finder to sync your iPhone, iPad, or iPod with your computer - Apple Support
If the issue persists, provide more details in a reply. Including the answers to the questions below will help:
- If there's an error seen, what does it say?
- Where did the music originate from?
- Have you been able to use Sync Library in the past or have you been able to sync music between the iMac and iPhone previously? If so, has anything changed?
Take care!