  The recent release of {\it Planck} data gives access to a full sky coverage
of the thermal Sunyaev-Zel'dovich (tSZ) effect and of the cosmic microwave
background (CMB) lensing potential ($\phi$).
  The cross-correlation of these two probes of the large-scale structures in
the Universe is a powerful tool for testing cosmological models, especially in
the context of the difference between galaxy clusters and CMB for the
best-fitting cosmological parameters.
  However, the tSZ effect maps are highly contaminated by cosmic infra-red
background (CIB) fluctuations.
  Unlike other astrophysical components, the spatial distribution of CIB varies
with frequency.
  Thus it cannot be completely removed from a tSZ Compton parameter map, which
is constructed from a linear combination of multiple frequency maps.
  We have estimated the contamination of the CIB-$\phi$ correlation in the
tSZ-$\phi$ power-spectrum.
  We considered linear combinations that reconstruct the tSZ Compton parameter
from {\it Planck} frequency maps.
  We conclude that even in an optimistic case, the CIB-$\phi$ contamination is
significant with respect to the tSZ-$\phi$ signal itself.
  Consequently, We stress that tSZ-$\phi$ analyses that are based on Compton
parameter maps are highly limited by the bias produced by CIB-$\phi$
contamination.
