This paper considers rational integer values of Kloosterman sums over finite fields of characteristicp> 3. We shall prove two main results. The first one is a congruence relation satisfied by possible integer values. One consequence is that there are no Kloosterman zeroes in the case of characteristicp> 3, which generalizes recent works by Shparlinski, Moisio, and Lisoněk on this subject. This, in turn, implies that there are no Dillon type bent functions in the casep> 3 , thus answering a question posed recently by Helleseth and Kholosha. Our other main result states that the Kloosterman sum obtains an integer value at a point if and only if the same sum lifted to any extension field remains an integer.