  In this paper we consider tree-level gauge invariant off-shell amplitudes
(Wilson line form factors) in $\mathcal{N}=4$ SYM. For the off-shell amplitudes
with one leg off-shell we present a conjecture for their Grassmannian integral
representation in spinor helicity, twistor and momentum twistor
parameterizations. The presented conjecture is successfully checked against
BCFW results for MHV$_n$, NMHV$_4$ and NMHV$_5$ off-shell amplitudes. We have
also verified that our Grassmannian integral representation correctly
reproduces soft (on-shell) limit for the off-shell gluon momentum. It is shown
that the (deformed) off-shell amplitude expressions could be also obtained
using quantum inverse scattering method for auxiliary $gl(4|4)$ super spin
chain.
