Welcome to Fish Forum!
Bettas do indeed have little nostrils, I think they're quite cute xD
How often do you change the tank's water? And how much? Bettas like clean water, and dirty water is often the case of most fin rot anyhow, keep your tank clean with regular water changes, and your betta will probably never catch fin rot again. Add a little aquarium salt in already if you haven't been doing that, it also does wonders to help ward off fin rot and bacterial infections.

The coloring could indeed be a fungal infection of some sort, or it may just be natural coloring. Does your betta have this coloring appearing anywhere else? I've run across several bettas who have white on the underside of their chin/face area, and they also have clear/white nostrils although the rest of their upper face is a different color.
However, if this is a recent addition to your betta's normal appearance, he may indeed have something (I don't think bettas go 'grey in the face' like dogs or cats, so I doubt its old age). If you can, trying an get a magnifying glass up next to the tank so you can get a better look at what it may be (pictures, if you have them, would be a great help to us in help you diagnose your fish) It may just be a localized fungal infection, at which point in time treating him for such would probably be a good idea. Could also be ick spots developing there, but that's unlikely that it would stay only in one spot, you'd notice the ich appearing on the rest of his body within several days anyhow. Some fish over produce their slime coat, and this can often cause a milky, filmy substance which sticks to certain parts of the fish more prominently than others, including the face region, eyes, and tail base. But that's not so much the nostrils themselves turning a different color, as the nostrils being covered with something that makes it look different.

 

Thanks for the reply!

He has only been in the new 5 gallon for 4 days now, and has had one small water change as per the T.C. directions. The fin rot looks to be under control, T.C. is strong stuff!

There appears to be no other "white-ish" areas on him. His body is black with blue scales, and red fins. It's just around the edges of the nostrils. The water is quite discoloured from the T.C. so I can't get a real good look until I replace the carbon filter. I am quite sure it is not ich. Perhaps he has always had these sort of white or clear nostrils and I have never noticed until I started examining him after he got the fin rot...

 

one thing you might want to take in consideration...you say you dont have a heater, as your home is warm anyway:
When I first got a Betta, I too had it in a small tank: 2.5 and its small size was perfect for room temp heating. However, the 10 gal tank I moved him into was too much water, and could never reach the temperature they need of 76-80 with just the room heating alone. I would defiantly [if you dont already have one] get a tank thermostat [either the glass kind that floats inside, or the exterior kind that stick to the tanks walls...typically removable].

That's the only way you'll be sure that your little guy is getting the proper heating it needs. Knowing your temp is right via thermostat is defiantly better then assumption via room temp. Cold water also would have a factor in your little guy's health.

As they're typically max price of $3 [unless you go all out with a fancy type], they should be within everyone's budget too.

good luck, and hope he's getting better.
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hae