I'm on the other side of the table right now in a really (coincidently) close situation. I'm interviewing everywhere, I have a set of skills that can't be found easily and we are in a cost saving situation.
Here's what I would hope my boss did if he ever found out.
Have a conversation
Invite the employee to your office and mention the rumours. You are obviously worried about this but make sure you don't look angry or scared or anything. So far, those are rumours. Treat them as such. If she says they are not true, accept the answer but ask the following question anyway.
The trick here is to be respectful of the sentiment of the employee towards the company. If she's leaving, there's a reason. It might be financial, it might be about "finding a new challenge" or go deeper. In my situation right now it's a mix of two things. I'm not paid enough and the cost saving mode makes me believe I might get cut sometime.
If you would get an offer somewhere, how long would your notice be?
It's a perfectly fine thing to ask. 2 weeks in my country is culturally acceptable but I already determined with a couple people in interview that if they want me, I'd like a month notice because I'm on important projects.
If she did get offers and tells you something that's culturally acceptable wherever you are, I suggest you take it and thank her for letting you know. Show your appreciation. My worst fear would be for an employer to try to sabotage my career to keep me where I am.
Else: Ask her if she could give you that culturally acceptable time so you can try to find someone if needed be
Either way, be clear that your intention is not to keep her here forever but just to be prepared.
You are a valuable member of our team and I'd rather have you stay then leave. Is there anything I can do to make your work here more appreciable?
Notice the way I formulated the question here. You show appreciation but you also want constructive criticism. You might not get any, but at least you'll learn something. Either she wants to leave for personal reasons or you can reassure her on some fronts or offer her something else. You might not be able to give her a better salary but maybe she'd be better in a different office or something else.
End the conversation with a reminder that you appreciate her time and skills and thank her for talking to you.
Last thing I'd want to know is that my boss would suddenly hate me or be all passive aggressive on me. I always get along with everyone I work with so if looking for a different job would make my workplace uncomfortable, I would be looking even more ferociously.