After an unsuccessful first PhD position in experimental research, how can I get accepted to another program?
If your relation with your current advisor is not good, you probably don't want to cite him/her as reference or ask them for recommendation. Usually one ask recommendation only from people we are sure they have a good opinion. Maybe someone else from your Master can give a better opinion of you. Or colleagues/postdoc/teaching assistant in your current position.

This does not mean you need to hide the current situation. It is better to introduce the problem when you say why you want to change position. And be super clear with the situation, if they ask you what was the problem, be open with it. If you are hesitant to speak, they will feel there is something shady and might become suspicious. Also you should not try to discredit your current supervisor, but you can try to explain why his/her expectations were not met.

If they still insist on contacting your current advisor, then let it be, there is not much more you can do.