Canned tuna, a lunchbox staple from the 20th century, is fighting to keep its spot in American cupboards.

Century-old tuna companies like StarKist Co., Bumble Bee Foods LLC and Chicken of the Sea International are trying to reboot demand for tuna fish—selling it in cans, pouches and kits with trendy flavors or as a healthy snack—as they seek to hold on to their dominance in a shrinking market.

“In...