This paper deals with the behavior of positive solutions to the following nonlocal polytropic filtration system $${u_t=(mid(u^{m_1})_x{mid}^{{p_1}^{-1}}(u^{m_1})_x)_x+u^{l_{11}}{{int_0}^a}v^{l_{12}}({xi},t)d{xi},;(x,t);in;[0,a]{ imes}(0,T),\{v_t=(mid(v^{m_2})_x{mid}^{{p_2}^{-1}}(v^{m_2})_x)_x+v^{l_{22}}{{int_0}^a}u^{l_{21}}({xi},t)d{xi},;(x,t);in;[0,a]{ imes}(0,T)}$$ with nonlinear boundary conditions $u_x{mid}{_{x=0}}=0$, $u_x{mid}{_{x=a}}=u^{q_{11}}u^{q_{12}}{mid}{_{x=a}}$, $v_x{mid}{_{x=0}}=0$, $v_x|{_{x=a}}=u^{q21}v^{q22}|{_{x=a}}$ and the initial data ($u_0$, $v_0$), where $m_1$, $m_2{geq}1$, $p_1$, $p_2$ > 1, $l_{11}$, $l_{12}$, $l_{21}$, $l_{22}$, $q_{11}$, $q_{12}$, $q_{21}$, $q_{22}$ > 0. Under appropriate hypotheses, the authors establish local theory of the solutions by a regularization method and prove that the solution either exists globally or blows up in finite time by using a comparison principle.