The alloy design is vital mission for eliminating the laser melting and additive manufacturing technologies defects. The low melting-point phases around the grain boundaries are considered as a source of the liquation cracks formation during laser melting process (LMP). In this work, a simple Al-Cu binary alloy with different Cu concentrations was selected as a model to understand the relation between low-melting point phases and liquation cracks formation during the LMP. In cast samples, most Al2Cu (θ phases) precipitate at the α-Al grain boundaries and during the LMP, the liquation cracks in the laser-melted zone (LMZ) initiated at the grain boundaries and propagated along the LMZ. Thus, homogenization annealing pre-laser melting at various times was done. The results showed that the cast Al-3.5Cu revealed high cracks susceptibility in the LMZ due to the presence of high amount of θ phases and during homogenization annealing the phases dissolved and the number of cracks significantly decreased. No cracks were formed in Al-7.5Cu at the cast and homogenized conditions due to the presence of many equilibrium eutectic θ phases, which heals the cracks (backfilling) during the solidification after the LMP. Liquation cracks susceptibility can be controlled by homogenization annealing.