Based on the information provided, it seems like there might be a setting on your iPhone that's causing this issue. Let's try a couple of troubleshooting steps before considering buying another pair of earbuds.
Firstly, check if there's any debris or dust in the headphone jack of your iPhone. Sometimes, particles can interfere with the proper connection and cause the Bluetooth prompt to appear. Take a close look and gently clean the jack using a soft, lint-free cloth or a cotton swab.
If the issue persists, it's worth checking your iPhone's audio output settings. Go to "Settings" on your iPhone, then tap on "Sounds & Haptics." Make sure that "Change with Buttons" is toggled off, as having it on can sometimes trigger the Bluetooth prompt.
If none of the above steps solve the issue, you can try resetting your iPhone's settings. Note that this won't erase your data but will restore your device's settings to default. Go to "Settings," tap on "General," then "Reset," and choose "Reset All Settings." After the reset is complete, test your wired earbuds again and see if the Bluetooth prompt disappears.
If you still encounter issues, it might be helpful to contact Apple Support for further assistance. To do so, simply select the "Support" link on the lower left-hand side of this forum's web page.
Here are a couple of Apple Support articles that may provide additional guidance:
1. Get help with the audio on your iPhone
2. If your headphones don't work with your iPhone, iPad, or iPod touch