You could use any bluetooth headset, the best chip to search for is
http://www.csr.com/products/csr8645 (but it doesn't really matter for ham radio)
However you need a special bluetooth interface for your radio if it does not have one, and here's the issue. A bluetooth module will transmit and receive at the same time only if it is set in headset or hands-free profile, and the only device that can do that is a cell phone or a custom device. Also, there is a catch because you would have to output and receive the ham radio's signals on the phone's headphone input/output.
https://en.wikipedia.org/wiki/List_of_Bluetooth_profiles#Hands-Free_Profile_.28HFP.29It's possible, but the main issue is software. In the past there have been bluetooth devices like the Jabra which converted an old phone to bluetooth and streamed directly back and forth. Nowadays that's not necessary unless one receives a call so one needs an application that is capable of setting the bluetooth headset in call mode and route the streams to the headphone output of a phone and with android that's not possible.
So unless you have a raspberry pi or maybe an orange pi zero with a USB Bluetooth module & sound card and you are willing to kill a few of your neurons trying to route the sound from one to another in Linux it's pretty hard to make an interface for a ham radio.
You could replace the raspberry pi or orange pi with a computer with either windows or Linux, but you still would have to write some software/configuration to do some magic and you could even replace the PTT switch with the accept call button on the headset.
If you are lucky you can just play around in Windows with the Headset Profile and listen to the bluetooth microphone on the sound card and listen to the microphone or line-in of the computer with the speaker of the headset.
Alternatively you can use an android/ios device to connect to a a radio remotely through the internet with an application that supports bluetooth headsets.