It seems you will have to file an unlawful detainer action against him to regain possession of the home. With a valid Notice of Trustee's Sale in hand and accompanying documents such as the deed, you can have him removed from your property.
Offering cash for keys is nice of you but if you have rightfully bought the property from an auction, you are under no obligation to negotiate with him. I recommend filing the unlawful detainer.
How long has he lived there? A few months, a few years, 10 years or more? Did the prior owners give him a lease or anything? I would gather all the deeds, title reports, etc. and find a local real estate attorney to review everything. There could be some issues here that need to be addressed.
You can file an unlawful detainer action to gain possession of the property if you recorded a Trustee's Deed Upon Sale. Unless the man has a valid claim and can prove that the foreclosure sale was somehow fraudulent, you are considered a "bona fide" purchaser which means that you are protected from any claims the man may have against the prior banks.
By posting on this site or answering/responding to questions does not create an attorney-client relationship and is intended to be an opinion only. My opinion is not intended to be a guarantee or promise of any outcome or result in your matter.