If it is your tree, on your property, then its likely to be your liability if it goes down and causes damage. You should make sure you have appropriate homeowners insurance coverage and notify the neighbor of the potential danger. And as soon as possible, you should get a tree-cutter out there.
If the branches of your tree extended over the boundary line beyond your property onto that of your neighbors, your neighbors would have the right to cut back the tree to the trunk. But your situation is different in two significant ways. First, you indicate that "[the tree] is not over the property line." This suggests that its branches are unreachable by your neighbor without a license to enter your land. Moreover, this suggests that your neighbor wants to take on the responsibility of sorting out any issues with your tree. However, the fact remains that even if the branches did extend over the boundary line, the neighbor would have only the right to remove them, and not a duty to do so.
It is true that you can call up your home insurance carrier and confirm that you have coverage for this type of incident. However, at best this will mean that you have put them on notice of the issue and no misfortunate event ever occurs. No misfortunate event other than a rise in your premiums at the time of renewal. On the other extreme, the tree falls and damages both property and person. In such an event, your carrier will likely tender a defense and pay damage for you up to your policy limits and then cancel your policy.
So, yes call the insurance company as a "band-aid," but then call an arborist to take care of your situation. It is quite likely that with a "haircut," the root system will be relieved of excess stress and as such the tree will not go down. Moreover, by initiating the conversation with your arborist via a letter or email that expresses your concern, if the tree still does fall down, you will be in a better position to evidentiarily demonstrate that you did what you could to eliminate a potentially negligent situation to both a court and your insurance company.
As a final note, make sure that your neighbors grants you a license to go upon their land for the purpose of taking down the tree if the arborist must do so to complete his job. Good Luck.