We study in this paper a general case of integer quadratic multi-knapsack problem (QMKP) where the objective function is non separable. An upper bound method is proposed for (QMKP) which is computed via two steps. The rst stage aims to rewrite (QMKP) into an equivalent mixed integer quadratic program (QPxy) where the objective function is separable, using Gauss decomposition of the quadratic terms matrix. We then suggest an original technique, we call roof linearization, to linearize (QPxy) so as to obtain a mixed linear program which optimal value provides an upper bound for (QPxy) and consequently for (QMKP). Preliminary computational ex- periments are conducted so as to assess that the proposed algorithm provides a tight upper bound in fast CPU times. Our method is compared with the LP-relaxation of (QMKP) and the LP-relaxation of (QPxy).