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This work discusses a general approach for preconditioning the block Jacobian matrix arising from the discretization and linearization of coupled multiphysics problem. The objective is to provide a fully algebraic framework that can be employed as a starting point for the development of specialized algorithms exploiting unique features of the specific problem at hand. The basic idea relies on approximately computing an operator able to decouple the different processes, which can then be solved independently one from the other. In this work, the decoupling operator is computed by extending the theory of block sparse approximate inverses. The proposed approach is implemented for two multiphysics applications, namely the simulation of a coupled poromechanical system and the mechanics of fractured media. The numerical results obtained in experiments taken from real-world examples are used to analyze and discuss the properties of the preconditioner. • A general framework is introduced for the block preconditioning of discrete systems arising from coupled multi-physics applications. • A decoupling operator is introduced by using the theory of block sparse approximate inverse preconditioners. • The framework is specialized for two applications: (i) coupled poromechanics, (ii) mechanics of fractured media. • Specific techniques for preserving the definiteness of the Schur complements are introduced for each application. • The preconditioners arising from the proposed general framework are used to solve real-world demanding applications. [ABSTRACT FROM AUTHOR]