The pseudogap phase occurring in cuprate and organic superconductors is analyzed based on the dynamical cluster approximation approach to the Hubbard model. In this method a cluster embedded in a self-consistent bath is studied. With increasing Coulomb repulsion, U, the antinodal point k = (pi,0) displays a gradual suppression of spectral density of states around the Fermi energy which is not observed at the nodal point k = (pi/2,pi/2). The opening of the antinodal pseudogap is found to be related to the internal structure of the cluster and the much weaker bath-cluster couplings at the antinodal than nodal point. The role played by internal cluster correlations is elucidated from a simple four-level model. The pseudogap can be understood in terms of destructive interference between different paths for electrons hopping between the cluster and the bath.