Can Fat Near the Inner Eye Area Be Removed?

Yes, the inner eye fat can be removed when it droops causing the area to be puffy. It tends to become puffier with allergies, altitude, air pressure, hormonal changes, and the aging process.
There are three compartments of fat in the lower lid, but the biggest compartment is the medial one which tends to be showing the bags the most prominently. If the bags are the only issue that needs to be addressed, that can be done through a transconjunctival approach with an incision on the inside of the eyelid. If there is excess skin on patients usually over 50 years of age, then a small pinch technique can be performed right at the lash line. This is usually closed with either fine dissolvable sutures or tissue glue.
 

Eyelid surgery, or blepharoplasty, is a beautiful procedure that yields excellent results and should be custom tailored in every aspect to your individual goals, needs, and anatomy. 
From your perspective as a patient, this surgery is very stealth with relatively simple recovery. From the standpoint of your surgeon this is a very intricate, detailed and fun surgery to perform where assessment and planning are everything!
Blepharoplasty has come a long way and become much more modern in recent years, focusing more on long-term results than just on taking away as much skin and fat as possible. This modern approach yields a much more natural look with much more satisfying long-term results. 
Blepharoplasty will address sagging skin, displaced, bulging fat and excess muscle. Occasionally areas around the eyes, like the eyebrows and sometimes even cheeks, need to be addressed to give a harmonious and comprehensive result.
There are non-surgical alternatives to blepharoplasty, including laser resurfacing, fillers, brow lifting and others, but which of those are applicable to you will be dependent, and a discussion with your surgeon can help delineate those options.
To ensure you are receiving the highest level of care, seek out a modernly trained, new-school dermatologic surgeon, oculoplastic surgeon, facial plastic surgeon or plastic surgeon who is board certified and fellowship trained in one of these "core four" cosmetic specialties. Membership in organizations like the American Academy of Facial Plastic and Reconstructive Surgery help to identify a highly trained surgeon.
Cameron Chesnut
#realself500 Physician

It sounds like you may need some fat removal and perhaps some excess skin removed as well. You will need to be evaluated in person by a facial plastic surgeon who has expertise in blepharoplasty.  This would still be an eyelid lift procedure.  I hope this helps.