Let $$\mathscr {F}$$ be a holomorphic foliation by curves defined in a neighborhood of 0 in $$\mathbb {C}^n$$ ( $$n\ge 2$$ ) having 0 as a weakly hyperbolic singularity. Let T be a positive $${dd^c}$$ -closed current directed by $$\mathscr {F}$$ which does not give mass to any of the n coordinate invariant hyperplanes $$\{z_j=0\}$$ for $$1\le j\le n.$$ Then we show that the Lelong number of T at 0 vanishes. Moreover, an application of this local result in the global context is given.