  The role of intrinsic $\bfk_\perp$ in inclusive and semi-inclusive Deep
Inelastic Scattering processes ($\ell p \to \ell h X$) is studied with exact
kinematics within QCD parton model at leading order; the dependence of the
unpolarized cross section on the azimuthal angle between the leptonic and the
hadron production planes (Cahn effect) is compared with data and used to
estimate the average values of $k_\perp$ both in quark distribution and
fragmentation functions. The resulting picture is applied to the description of
the weighted single spin asymmetry $A_{UT}^{\sin(\phi_\pi - \phi_S)}$ recently
measured by the HERMES collaboration at DESY; this allows to extract some
simple models for the quark Sivers functions. These are compared with the
Sivers functions which succeed in describing the data on transverse single spin
asymmetries in $\pup p \to \pi X$ processes; the two sets of functions are not
inconsistent. The extracted Sivers functions give predictions for the COMPASS
measurement of $A_{UT}^{\sin(\phi_\pi - \phi_S)}$ in agreement with recent
preliminary data, while their contribution to HERMES $A_{UL}^{\sin\phi_\pi}$ is
computed and found to be small. Predictions for $A_{UT}^{\sin(\phi_K -
\phi_S)}$ for kaon production at HERMES are also given.
