Let \(T\) be a positive plurisubharmonic (\(i\partial\overline\partial T\geq 0\)) or plurisuperharmonic (\(i\partial\overline\partial T\leq 0\)) current on a neighborhood of \(0\) in \({\mathbb C}^n\). One says that \(T\) has a tangent cone at \(0\) if the weak limit of the family of its homothetic currents exists. In this paper, the authors give a sufficient condition on the projective mass of \(T\), guaranteeing the existence of a tangent cone of \(T\). The proof frequently uses the Lelong-Jensen formula. When \(T\) is closed, the authors recover the result given in [M. Blel et al., Ark. Mat. 28, No. 2, 231–248 (1990; Zbl 0724.32005)]. Moreover, some estimates on the the growth of the Lelong function associated to a plurisubharmonic or plurisuperbharmonic current, are obtained in order to prove the existence of the strict transform of such currents.