The Root of the Behavior
When your dog stares imploringly into your eyes, he most definitely has a message for you. Staring is a means of gaining your undivided attention. You are the center of your dog's world, and his most cherished "possession." For you to engage with him is the most powerful reinforcer he knows. However, sometimes your dog wants more than just your attention. In your role as the giver of all good things, you possess the sole capability to give Fido what he wants and what he most often wants is food. Yes, dogs are opportunists. They know precisely what they need to do in order to get the result they want from you. Dogs carefully observe past behaviors and make note of patterns. If following you around and delivering intense stares has led you directly to the cookie jar in the past, Fido is going to keep trotting out this behavior until it no longer yields the result he wants any more.
Encouraging the Behavior
But sometimes our dogs are communicating things to us that go far beyond simple wants to actual physical needs of importance. If your dog does not have free access to use the bathroom when he needs to, his staring might be his attempt to let you know that he is desperate to go outside to do his business. This type of staring is generally accompanied with pacing, pawing, and a frenetic behavior that tells you that something is wrong...or about to be wrong if you don't heed his warning signals. Some dogs attempt to stare into the eyes of their owners because they are seeking direction. This is very common behavior exhibited during training sessions especially if you are working on exercises that involve shaping. Shaping is a training tool where you offer clicks, food rewards, and praise when your dog offers you a behavior that you are seeking. It is a means of giving your dog a choice in the training process and is helpful to encourage voluntary participation. Some dogs offer behaviors quite readily while others become perplexed by the process. It is then that a dog would look to his owner for guidance. This is a very positive thing and should be encouraged.
Other Solutions and Considerations
To do this, it is important to each Fido the "off" command. "Off" is a learned skill that is an important part of every canine's "trick" repertoire or should be. To teach the "off" command, you will need a clicker and some yummy treats. When Fido gets a little too close for comfort, you can then use your treat to lure him into the position where you would like him to rest while saying the word "Off!" Be certain to state the word in a firm, authoritative voice. It is not a question; it is a command, and compliance is expected. When your dog performs the required behavior, you can then click him, provide him with his treat, and lavish praise on him. It will take some time for Fido to truly grasp this concept then consistently deliver it when asked for it, but it will come over time.