  We describe the search for Lyman-break galaxies (LBGs) near the
sub-millimeter bright starburst galaxy HFLS3 at $z$$=$6.34 and a study on the
environment of this massive galaxy during the end of reionization.We performed
two independent selections of LBGs on images obtained with the \textit{Gran
Telescopio Canarias} (GTC) and the \textit{Hubble Space Telescope} (HST) by
combining non-detections in bands blueward of the Lyman-break and color
selection. A total of 10 objects fulfilling the LBG selection criteria at
$z$$>$5.5 were selected over the 4.54 and 55.5 arcmin$^2$ covered by our HST
and GTC images, respectively. The photometric redshift, UV luminosity, and the
star-formation rate of these sources were estimated with models of their
spectral energy distribution. These $z$$\sim$6 candidates have physical
properties and number densities in agreement with previous results. The UV
luminosity function at $z$$\sim$6 and a Voronoi tessellation analysis of this
field shows no strong evidence for an overdensity of relatively bright objects
(m$_{F105W}$$<$25.9) associated with \textit{HFLS3}. However, the over-density
parameter deduced from this field and the surface density of objects can not
excluded definitively the LBG over-density hypothesis. Moreover we identified
three faint objects at less than three arcseconds from \textit{HFLS3} with
color consistent with those expected for $z$$\sim$6 galaxies. Deeper data are
needed to confirm their redshifts and to study their association with
\textit{HFLS3} and the galaxy merger that may be responsible for the massive
starburst.
