Yes, it's possible. The idea of inverting a were-creature into a "creature-were" isn't new and has been done in various editions of D&D. There are various monsters and "PC" races (and because this is a game of imagination, new ones can — and are, often created): hengeyokai, jackalweres (not werejackals!), wolfweres, kitsune, and many others are all shapeshifters of some kind that are natively animals but can assume human form, and which have been implemented in some edition of D&D in some form that could allow creating one as a character. The concept is well-represented in D&D!
I can't give you a definitive list and won't try (and it would quickly be incomplete even if I did), but the main point is demonstrated: yes, you can make a "were-human" character.
Depending on the edition and DM that you would be trying this with, your options may be more or less constrained. Some of them are monsters, and creating such a character would require an edition and DM that allow and have rules for creating PCs out of "monster" races; some of them are PC races but are found in "non-core" books that may not be permitted at a given table.
So yes, it's possible and there's support for it in D&D taken as a whole, and the ability to do it is actually pretty common across editions even if the "how" varies considerably. The details of each way to get to this goal are as diverse as the species that can do it and as there are editions of D&D, so it would heavily depend on the details of your specific future play situation if and when you attempt to create such a character.
(Because mechanics vary considerably, they can't really be addressed properly without a specific game, DM, and group that you're actually making a character for. Should you find yourself in such a situation, that would make a great follow-up question.)