Be careful, there is something wrong with these Gigaset E720s. All of our Bluetooth headsets connect without any problems (which answers your question: you can choose any Bluetooth headset). However, the listening volume is much too low when using a Bluetooth headset (output volume of the Gigaset E720 device much too low?), even if the Bluetooth headset volume is set to maximum in the phone. You can hardly understand the person you are talking to, even in a quiet room. It's not because of the Bluetooth headsets (I tried 3 different ones from different manufacturers), they work perfectly with my iPhones.
We don't have a "Monday device" either, we bought 2 Gigaset E720s and it's the same problem with both.
We have complained several times to Siemens, they always answer, but completely unusable (so in the sense: we are not aware of these problems). I have equipped my family with these Gigasets E720 and with relatively expensive Jabra headphones and am frustrated to the max. If it works for you, please feel free to let me know.