Recently, Xiang and Ouyang defined a (commutative unital) ring to be Nil-coherent if each finitely generated ideal of that is contained in Nil is a finitely presented -module. We define and study Nil-coherent modules and special Nil-coherent modules over any ring. These properties are characterized and their basic properties are established. Any coherent ring is a special Nil-coherent ring and any special Nil-coherent ring is a Nil-coherent ring, but neither of these statements has a valid converse. Any reduced ring is a special Nil-coherent ring (regardless of whether it is coherent). Several examples of Nil-coherent rings that are not special Nil-coherent rings are obtained as byproducts of our study of the transfer of the Nil-coherent and the special Nil-coherent properties to trivial ring extensions and amalgamated algebras.