However, if this is a pile type rug, I would get some carpet tape from a home center, then go to an excluded place in your home like a closet that has the same carpet and cut out a square barely the size of the chew hole. Then go to the chew hole and square off this bite hole to the same size as the patch square. Then lift up the chew holed carpet enough to insert the carpet tape under the entire chew hole. Now, insert the new patch and then get your iron and set on medium or so.
Lay an old pair of faded jeans over the hole and then with the iron, iron the hole area so that the glue bead will melt and hold the patch to the chew holed carpet. Check it periodically to make sure that you are not shining the carpet.
Normally you iron the underneath side of the carpet, but in this case it is impossible, unless you can pull the carpet up and that you are close to the tack strip.