This paper describes the output voltage regulation control for an interleaved connected to a direct current (DC) microgrid considering bidirectional current flows. The proposed controller is based on an interconnection and damping passivity-based control (IDA-PBC) approach with integral action that regulates the output voltage profile at its assigned reference. This approach designs a control law via nonlinear feedback that ensures asymptotic stability in a closed-loop in the sense of Lyapunov. Moreover, the IDA-PBC design adds an integral gain to eliminate the possible tracking errors in steady-state conditions. Numerical simulations in the Piecewise Linear Electrical Circuit Simulation (PLECS) package for MATLAB/Simulink demonstrate that the effectiveness of the proposed controller is assessed and compared with a conventional proportional-integral controller under different scenarios considering strong variations in the current injected/absorbed by the DC microgrid.