Advice for propagating persimmon?

I have a 2 year old Nikita's Gift hybrid persimmon tree in my yard that has a number of small shoots growing from its root system. I was wondering whether or not these would produce fruit-bearing trees, given that the original specimen I purchased was (I believe) grafted by the nursery onto American persimmon root stock, and the shoots are growing from these roots. If so, what would be the correct way to go about replanting them? Do they need to be grafted? They are getting fairly large and I don't want to wait much longer if they need replanting. Thank you!

The root suckers, if not removed, will take a lot of energy out of your cultivar.  Cut the root sucker so it is separated from the host tree.  It is likely a D. Virginiana.  Let them establish their own root system and when established transplant and graft some interesting cultivars on to them.  You will enjoy the cultivars more than the small and seedy Native Persimmon.