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By constructing a special cone and using fixed point index theory, this paper investigates the existence of positive solutions of singular superlinear coupled integral boundary value problems for differential systems $$\left\{ \begin{array}{ll} -x''(t)=f_1(t,x(t),y(t)),\ \ t\in (0,1),&\\ -y''(t)=f_2(t,x(t),y(t)),\ \ t\in (0,1),&\\ x(0)=y(0)=0, x(1)=\alpha[y], y(1)=\beta[x],& \end{array} \right.$$ where $\alpha[x], \beta[x]$ are bounded linear functionals on $C[0,1]$ given by $$\alpha[x]=\int_0^1x(t)dA(t), beta[x]=\int_0^1x(t)dB(t)$$ with $A,B$ functions of bounded variation with positive measures.