We deal with the existence of positive radial solutions concentrating on spheres for the following class of elliptic system $$large(S) hfill{400} {array{-{varepsilon}^2{Delta}u+V_1(x)u=K(x)Q_u(u,v);in;mathbb{R}^N,\-{varepsilon}^2{Delta}v+V_2(x)v=K(x)Q_v(u,v);in;mathbb{R}^N,\u,v{in}W^{1,2}(mathbb{R}^N),;u,v>0;in;mathbb{R}^N,}$$ where ${varepsilon}$ is a small positive parameter; $V_1$, $V_2{in}C^0(mathbb{R}^N,[0,{infty}))$ and $K{in}C^0(mathbb{R}^N,[0,{infty}))$ are radially symmetric potentials; Q is a $(p+1)$-homogeneous function and p is subcritical, that is, 1 < $p$ < $2^*-1$, where $2^*=2N/(N-2)$ is the critical Sobolev exponent for $N{geq}3$.