Elk are exotics in Texas

From regs:
Exotic animal refers to grass-eating or plant-eating, single-hoofed or cloven-hoofed mammals that are not indigenous or native to Texas and are known as ungulates, including animals from the deer and antelope families that landowners have introduced into this state. Includes, but is not limited to feral hog, Aoudad sheep, Axis deer, Elk, Sika deer, Fallow deer, Blackbuck antelope, Nilgai antelope, and Russian boar. Exotic fowl refers to any avian species that is not indigenous to this state, including ratites (emu, ostrich, rhea, cassowary, etc.).

There are no state bag or possession limits or closed seasons on exotic animals or fowl on private property. It is against the law to:

Hunt an exotic without a valid hunting license.
Hunt an exotic on a public road or right-of-way.
Hunt an exotic without the landowner's permission.
Possess an exotic or the carcass of an exotic without the owner's consent.