We give conditions on a pair of Banach spaces X and Y, under which (*) each operator from X to Y, whose second adjoined member compactly factors through the space l^p, 1≤p≤+∞, itself compactly factors through l^p. The conditions are as follows: either the space X^*, or the space Y^*** possesses the Grothendieck approximation property. Leaving the corresponding question for parameters p>1, p≠2, still open we show that for p=1 the conditions are essential: there exist Banach spaces with the bases for which the assertion (*) is invalid for the case of l¹-factoring operators.