The Collaborative International Dictionary of English v.0.48:

djinnee \djin"nee\ djinni \djin"ni\, djinny
\djin"ny\(j[i^]n"n[=e]), n.; pl. djinn (j[i^]n) or djinns
   (j[i^]nz).
   A spirit believed by Muslims to inhabit the earth and
   influence mankind by appearing in the form of humans or
   animals. Same as djinni and Jinnee. See Jinnee, Jinn.

   Syn: genie, jinn, jinni, jinnee, djinn, djinni.
        [1913 Webster + WordNet 1.5]

WordNet (r) 3.0 (2006):

djinni
    n 1: (Islam) an invisible spirit mentioned in the Koran and
         believed by Muslims to inhabit the earth and influence
         mankind by appearing in the form of humans or animals [syn:
         genie, jinni, jinnee, djinni, djinny, djinn]