Asking an employee about any private conversation puts up a bunch of red flags. Even if the question was simply asked in conversation it is just about guaranteed to put an employee on the defensive. And, a defensive employee is never going to be operating at maximum productivity or loyalty.
My advice is that you remain calm, sit down with the questioner and voice your concerns.
Here is my 10-step plan for holding a difficult conversation.
1. Prepare. Make some notes about the situation and your feelings. Write about where you are, where you want to be and how you might get there.
2. Come to the table and stay there. The other side will come if your message is "I truly want to find a solution that works for both of us." If you cannot carry the message, find someone who can intervene on your behalf and get you both to the table.
3. Set the stage. Sit down at a time when you are both clear headed and able to give this important conversation the time and energy it deserves. Make sure that the environment is private.
4. Speak from the heart. Do not point fingers of blame. Instead focus on finding a solution that works for both of you. This is collaboration.
5. Listen, listen, listen. Listen as if you are an outside observer with no prior knowledge of the situation. Twenty years in the mediation business has taught me that there are at least two sides to every story. You may be very surprised when you hear the rest of the story.
6. Give yourselves time to think, process the information, and cool down.
7. Define the emotions. Under almost every human conflict someone feels dismissed, discounted, disenfranchised or disrespected. Sometimes, just defining that emotion and realizing that both of us feel the same way--devalued--is enough to resolve the dispute.
8. Be willing to apologize. The closer the relationship, the more likely you are to have stepped on each other's toes. If you cannot bring yourself to apologize for anything specific at least apologize for the distress that the other side has been living with and anything s/he believes you did to contribute to it.
9. Don't leave conflicts unresolved. An agreement to disagree is resolution. Leaving the conflict open sets you up for future fights.
10. If all else fails, hire a professional to help you. Often an outside opinion sheds light on your blind spots and helps reach an agreement. Consider bringing in a mediator when the relationship is important.