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Let $q$ be a positive integer, $\chi$ a nontrivial character mod $q$,$\mathcal{I}$ an interval of length not exceeding $q.$ In this paper we shallstudy the character sum analogue of the well-known Kloostermansum,\[\sum_{\substack{a\in\mathcal{I}  \gcd(a,q)=1}}\chi(ma+n\overline{a}),\] where $\overline{a}$ is themultiplicative inverse of $a\bmod q$. The mean square values and bilinear formsfor such sums are proved.

Mean values of character sums analogue of Kloosterman sums