  We report photometric observations of the trans-Neptunian object
2004~TT$_{357}$ obtained in 2015 and 2017 using the 4.3~m Lowell's Discovery
Channel Telescope. We derive a rotational period of 7.79$\pm$0.01~h and a
peak-to-peak lightcurve amplitude of 0.76$\pm$0.03~mag. 2004 TT$_{357}$
displays a large variability that can be explained by a very elongated single
object or can be due to a contact/close binary. The most likely scenario is
that 2004 TT$_{357}$ is a contact binary. If it is in hydrostatic equilibrium,
we find that the lightcurve can be explained by a system with a mass ratio
q$_{min}$=0.45$\pm$0.05, and a density of $\rho_{min}$=2g cm$^{-3}$, or less
likely a system with q$_{max}$=0.8$\pm$0.05, and $\rho_{max}$=5g cm$^{-3}$.
Considering a single triaxial ellipsoid in hydrostatic equilibrium, we derive a
lower limit to the density of 0.78g cm$^{-3}$, and an elongation (a/b) of 2.01
assuming an equatorial view. From Hubble Space Telescope data, we report no
resolved companion orbiting 2004 TT$_{357}$. Despite an expected high fraction
of contact binaries in the trans-Neptunian belt, 2001 QG$_{298}$ is the unique
confirmed contact binary in the trans-Neptunian belt, and 2004 TT$_{357}$ is
only the second candidate to this class of systems, with 2003 SQ$_{317}$.
