Local positive reachability of positive nonlinear continuous-time systems is investigated. It is shown that the system is locally positively reachable from the origin if and only if in some neighborhood of the origin, for each subface of the nonnegative cone of the state space there is a control value such that the vector field corresponding to this control value is tangent to the subface and nonzero. This is a natural generalization of a characterization of positive reachability of positive linear continuous-time systems. Contrary to discrete-time systems, positive reachability of continuous-time systems requires that very restrictive conditions be satisfied.