I would get some clear silicone caulk. Tape around the tear with masking tape, so extra caulk that squeezes out of the repair won't get all over the rest of the "tile" face.
Carefully pry up the places where the tear has occurred with a knife blade, inject the caulk into the spaces. Try to get some caulk under the final decorative color layer too, if possible. Then lay everything back out as close to the untorn state as possible.
Lay a bit of wax paper, cling wrap, or plastic sheet over the top of the tear. Lay a flat board over the plastic. Lay heavy weight on the board.
Let the caulk dry for 2 or 3 times its curing time. When the board and plastic are removed, the tear and caulk should be squeezed flat again. It may not look 100% factory, but it will at least be sealed shut.
Carefully pry up the masking tape. and clean off excess caulk that squeezed out.
This tear may have happened because something was dragged across the floor. Be careful about dragging things on vinyl floors.