The notion of bounded pseudo-amenability was introduced by Y. Choi and et al. [CGZ]. In this paper, similarly, we define bounded pseudo-contractibility and then investigate bounded pseudoamenability and contractibility of various classes of Banach algebras including ones related to locally compact groups and discrete semigroups. We also introduce a multiplier bounded version of approximate biprojectivity for Banach algebras and determine its relation to bounded pseudo-amenability and contractibility.