They can enquire, you don't have to answer. They can use that information any way they see fit including sharing it with your supervisor. It would be illegal for them to fire you for it though and it would also be illegal for the supervisor to give you less favourable treatment.

For future reference, you should not give prospective employers permission to contact your current employers if the option is presented to you. Because it creates awkwardness like this.

You've accepted the job offer, so your new company probably thought it was ok to contact your current employers.

I think the biggest issue you need to worry about is getting that new contract and agreeing a start date. Don't worry about your existing company and what they might think.