Ask Mr. Smarty Plants

 Sideroxylon lanuginosum (Gum bumelia) is not a very big tree ordinarily and is considered a "pest tree" in some localities. However, we respect the fact that you would like to preserve your old tree if you can. This member of the Mr. Smarty Plants Team grew up in West Texas, where there were few trees, with or without holes, so we have no personal experience in the matter.

We have to be honest, our first reaction to this question was "You're kidding, right?" But we did some research and discovered that this was once a common practice. Read this paper on Tree Surgery and the Concrete Cure. However, it was published by Concrete International, so it might not be totally impartial.

Moving on, we found some more articles that addressed the subject:

Previous Mr. Smarty Plants answer  on filling hole in tree with concrete.

From Gardening Know How Filling Holes in Tree Trunks

Califonia Garden Clubs, Inc. Do Not Fill Tree Cavities with Concrete

and, finally, the one we think makes the most sense -

Texas !&M AgriLIFE Extension Tree Care Kit

Whether  you want to do this yourself or engage a licensed arborist to make repairs or recommendations, you are the one that will have to make the decision. It will be necessary to weigh the costs and difficulties of treating the hole against the possibility that the tree will not survive much longer, no matter what you do.