  We generalize Kudryavtseva and Mazorchuk's concept of canonical form of
elements in Kiselman's semigroups to the setting of a Hecke-Kiselman monoid
$\mathbf{HK}_\Gamma$ associated with a simple oriented graph $\Gamma$. We use
confluence properties to associate with each element in $\mathbf{HK}_\Gamma$ a
normal form; normal forms are not unique, and we show that they can be obtained
from each other by a sequence of elementary commutations. We finally describe a
general procedure to recover a (unique) lexicographically minimal normal form.
