Let $Q_1,\cdots ,Q_r$ be quadratic forms with real coefficients. We prove that for any $ϵ\>0$ the system of inequalities $|Q_1\left(x\right)|\<ϵ,\cdots ,|Q_r\left(x\right)|\<ϵ$ has a nonzero integer solution, provided that the system $Q_1\left(x\right)=0,\cdots ,Q_r\left(x\right)=0$ has a nonsingular real solution and all forms in the real pencil generated by $Q_1,\cdots ,Q_r$ are irrational and have rank $\>8r$.