That would be grammatically correct but it would not capture the fact that the manager who should conduct the interview is new. Moreover, current seems redundant. It would be odd for the previous manager, who no longer works in the organization, to conduct an interview.
It depends what you want to say. With whom do you want to make the contrast?
The current manager should interview this man.
This seems rather odd. Who else are you supposing might do it? By using current, you are essentially opposing the manager to all former and future managers.
The new manager should interview this man.
This makes more sense. It could be meant as opposed to the former manager.