Coyotes are curious animals. Those that are more daring are likely to come to the fence to check out the dogs. To get a better sense of your situation, I have a couple of questions: Have the coyotes made an attempt to get over or under the fences? Have you ever had a dog attacked by coyotes?
Follow up after further communication: The reason I asked those questions is that the IDNR will typically not issue property owners a removal permit unless wildlife have caused property damage or are a health concern. If the coyotes were damaging your fences or had attacked a domestic animal, you'd have a case for getting a permit. A Nuisance Wildlife Control Operator (NWCO) might be willing to trap the coyotes for you (they charge a fee for the service). However, they would most likely euthanize the animals. (IDNR prefers for NWCO to handle coyote issues because they have more experience than the typical property owner in handling wildlife).
Unfortunately, relocating coyotes is not a viable option. They are very smart animals and are hard to trap, which increases time and money costs associated with live-trapping. But, even if a person managed to live-trap them, there really is no place to take them. Coyotes are a very common species in Illinois. And they have relatively large home ranges. Which means that if you trap a coyote and release it somewhere else, you have almost certainly just released it into the territory of another coyote. Now two coyotes have to fight over the food, water, shelter, and breeding space available in that habitat. What typically happens is that the released coyote is driven out of the area by the established coyote. And they often get hit by cars trying to make it back to their own territory. Releasing coyotes into a new area can also spread diseases which can have negative impacts on the coyote population.
While I understand your concern about having the coyotes so close, you probably don't need to worry about the coyotes causing a problem. They are curious animals. They want to check out the other animals in the neighborhood. If they are staying back from the fence, I wouldn't be too concerned. Coyotes live in urban environments. Even cities like Chicago have populations of coyotes! They are very helpful in keeping populations of rodents and rabbits under control.
To deter the coyotes from coming close, make sure all garbage and pet food is securely stored. Food (human or pet) that is left outside can attract mice and other rodents, and that will attract coyotes. If you notice the coyotes close to the fence you can try shouting at them or shaking a can with rocks in it. Spraying them with water from a garden hose may also work to scare them away. For more information about coyotes you could check out: http://web.extension.uiuc.edu/wildlife/directory_show.cfm?species=coyote
