Preheat a panini maker to medium heat.
Open tuna can and drain water. Chopped tuna pieces finely and place in a small bowl.
Add celery, pepper, salt, mustard, and mayonnaise to the tuna and combine.
Lay out the tortillas. Place a small amount of additional mustard in the middle of each tortilla (optional). Lay 1 ounce of cheese on each tortilla.
Divide the tuna salad mixture evenly between the two tortillas and fold to create a wrap (burrito style).
Place both wraps on the preheated panini maker and close the top. Let it cook until the cheese melts, the tuna is warm, and there are brown toast marks on the tortilla - about 5 minutes.
Remove from the heat, slice in half diagonally, serve with chips, crackers, or a side salad.