The 27-year-old spent three and a half years with Huddersfield and helped the Yorkshire club win promotion to the top flight last season.
Burnley manager Sean Dyche has brought in two other strikers during the transfer window with Chris Wood coming from Leeds United and Jon Walters arriving from Stoke City.
Manager Sean Dyche said: "We welcome Nahki as another striking option and a player that is different to those we have.
"He holds a different-skill set and therefore we feel can only enhance our attacking options in due course."
Wells is currently recovering from an ankle injury and said it would be "two or three weeks" before he is back to fitness.
(Reporting by Simon Evans Editing by Jeremy Gaunt.)
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