Every workplace will have a rumor mill. I don't care what you think, it's human nature to share experiences and to try to seem "in the know". Some people are more aware and skilled about how rumors start. They can help or they can hurt, but the fact is they carry a lot of weight with all employees. Now lets break this down a bit more for you.
Positive rumors bring change you want or needs where negative impacts productivity and brings down your workers. Both are stresses on your employees, but for different reasons. Lets say the latest rumor is so and so was given a bonus. For what now comes the positive or negative weight. If the rumor is because they were friends with someone the worry is "well who am I friends with? I'm not friends with the right people." You can see the dive. If it was oh they exceeded their goal. It's a positive. The worry is "Can I do that? If I do I will get that bonus"
No we come to the corporate need to control and stop rumors. You can't you want to, but you can't. Yes you can stop trade secrets and the release of privileged information, but you will never stop people from sharing their thoughts. Counseling and Firing an employee for their opinion of what an inept manager you are can and does lead to far more criticism publicly.
The best way I've seen this handled was talking to the employee.. The manger was very direct and asked what the problem was. The person voiced their opinion and off they went back to work. (more happened but that's the short story) I've seen it handled in the most drawn out painful way Employee is called in to confront the accusation of rumors spread about the employee's and told not to have rumors again. Yes that is what they were told "you there stop giving people reason to talk behind your back or else we will believe them over you"
The second one, educed to the statement should indicate the problem. As a manager your employees need to trust you. Telling them others have more weight then them is not a good productivity tool. If you already know there is a problem blaming them isn't going to work anyway. What is more is the person who makes it a point to "nicely" inform you. Don't trust that person, ever. It's that simple. Like it or a not a team doesn't like to tattle tale on one another. The only reasons, short of a crime, is because they are hoping to get something or they need reassurance you trust them. It's an information peace offering. Don't fall for it.
Rumors are a good chance to show you are an effective manager. You are listening, you care and you are active. Never approach the subject of the rumor with "well I hear you are..." it's a quick way to look like a kid saying I know the truth and no one wants that. Better to just ask, not draw it out, not make a big deal. You my find it was just a bad day and they said something, you employee humans and that's what happens.