  We report variational Monte Carlo calculations for the spin-$\frac{1}{2}$
Heisenberg model on the kagome lattice in the presence of both nearest-neighbor
$J_1$ and next-nearest-neighbor $J_2$ antiferromagnetic superexchange
couplings. Our approach is based upon Gutzwiller projected fermionic states
that represent a flexible tool to describe quantum spin liquids with different
properties (e.g., gapless and gapped). We show that, on finite clusters, a
gapped $\mathbb{Z}_{2}$ spin liquid can be stabilized in the presence of a
finite $J_2$ superexchange, with a substantial energy gain with respect to the
gapless $U(1)$ Dirac spin liquid. However, this energy gain vanishes in the
thermodynamic limit, implying that, at least within this approach, the $U(1)$
Dirac spin liquid remains stable in a relatively large region of the phase
diagram. For $J_2/J_1 \gtrsim 0.3$, we find that a magnetically ordered state
with ${\bf q}={\bf 0}$ overcomes the magnetically disordered wave functions,
suggesting the end of the putative gapless spin-liquid phase.
