Nullnorms with an annihilator a in any point of a bounded lattice are generalizations and unifications of t-norms and t-conorms. This study continues to investigate the construction of nullnorms on bounded lattices. We propose some methods to construct nullnorms derived from t-norms and t-conorms on bounded lattices, where some sufficient and necessary conditions on theirs annihilator are required. As a by-product of these constructions, idempotent nullnorms on bounded lattices are obtained. Further, we provide some illustrative examples of the new classes of nullnorms (idempotent nullnorms) on bounded lattices. (C) 2019 Elsevier Inc. All rights reserved.