Would the concentration of oxygen around fire increase as we get close to the fire (from the outside)
A fire has no way to directly "draw in" oxygen for it to burn. It consumes what oxygen is in the immediate vicinity of the flame, depleting the air in oxygen compared to the concentration of oxygen in air further away. This sets up a concentration gradient such that oxygen will diffuse into the depleted region near the fire.  

An even more important process than diffusion of oxygen into depleted air is convection driven by the heat of the fire. Oxygen-poor air that has been in the fire gets hot and rises, drawing fresh air into the base of the fire. Note that this still isn't oxygen being drawn to the fire, but a mass of fresh air flowing into the lower pressure area at the fire resulting from convection.  

The bottom line is that a fire consumes oxygen, so the region around the fire will be slightly depleted in oxygen, not enriched as you suggest.