Interfaces should define public procurement, so it makes no sense to have a private member in it.
At least so far. There are proposals probably for C# 8 interfaces to allow implementations and then it would make sense to have private members, although the name interface starts to lose meaning.
At the moment you should simply ignore the private member, if private is implementation detail, then let the class decide on this. When the interface behaves like a trait the private member will be detail of the interface (in C# 8 there are still limitations in this, but already advanced, who knows more forward has more, has no forecast). You probably have a protected member, then you can do what you want, but there are controversies about using a protected member, even more in an interface, even if it is no longer an interface.
public interface IBar {
string Id { get; }
}
class Foo : IBar {
public string Name { get; private set; }
}
I put in the Github for future reference.