Is this for another job with your same employer? If so, this is the normal practice. They have to make sure that you are good-to-go and performing at an acceptable level in order to be considered for a transfer to another position. If it is for an external (different employer) job, it is unusual for to force the issue by requiring contact with your current supervisor... usually prospective employers are more than willing to keep everything on the 'down low' until you have the job.
All you can do is meet with your supervisor and explain why you are thinking about leaving. At some point, you will have to come clean & let your Sup know you're looking. It is not the end of the world. Job mobility is a fact of life with nurses. Never burn your bridges because you don't know what the future holds. In many organizations, managers/supervisors are limited in the type of references that they are permitted to provide to external requests.
Good luck on your job hunt.