Recently, Paiva et al. introduced the concept of quasi-overlap functions on bounded lattices and investigated some vital properties of them. In this paper, we continue consider this research topic and focus on the constructions of quasi-overlap functions along with their generalized forms on partially ordered sets. To be specific, firstly, we generalize the truth values set of quasi-overlap functions from bounded lattices to bounded partially ordered sets and introduce the notions of 0_P-quasi-overlap functions, 1_P-quasi-overlap functions and 0_P,1_P-quasi-overlap functions on any bounded partially ordered setPby considering the weaker boundary conditions than the quasi-overlap functions onP. Secondly, we give the constructions of quasi-overlap functions, 0_P-quasi-overlap functions, 1_P-quasi-overlap functions and 0_P,1_P-quasi-overlap functions on any partially ordered setPvia the so-called Galoiss-connections and 0,1-homomorphisms, 1-homomorphisms, 0-homomorphisms and ord-homomorphisms, respectively. In particular, we prove that those constructions contain the methods of extending the known quasi-overlap functions, 0_P-quasi-overlap functions, 1_P-quasi-overlap functions and 0_P,1_P-quasi-overlap functions from any partially ordered setPto any other partially ordered sets. Finally, we show that those extensions maintain some basic properties of the known quasi-overlap functions, 0_P-quasi-overlap functions, 1_P-quasi-overlap functions and 0_P,1_P-quasi-overlap functions onP, such as, idempotent, Archimedean property and cancellation law.