Wide nose job
Botox for dynamic alar flaring
From your statement the problem appears to be due to stong dialator and levator muscles of nose.
If you are happy with the appearance of the nostrils when you are not lauging Botox injection to the muscle of the nose to deactivate them may be an option. (Zain Kadri, MD, Los Angeles Facial Plastic Surgeon)
Wide nostrils
With proper technique, widened alae can be narrowed and still look very natural. Sometimes, other areas of your nose will need slight correction at the same time, to provide balance.
The key is to find a rhinoplasty surgeon who has experience and is board certified. (Dustin L. Reid, MD, Great Neck Facial Plastic Surgeon)
Rhinoplasty can narrow a wide nose naturally with an ALar Base Reduction
The part of the nose you describe is the fleshy section of the lower third which is the nostril skin. No cartilage exists here.
The way to reduce its width is by removing a few millimeters of skin with a procedure called an alar base reduction with an incision made where the nostril meets the cheek.
When executed well it looks natural and the scars are not apparent. Ask your surgeon how many of these he or she performs a year as this can be a tricky technique. (Andrew Jacono, MD, Dallas Facial Plastic Surgeon)