The effect of nonperiodic boundary conditions on decaying two-dimensional magnetohydrodynamic turbulence is investigated. A circular domain with no-slip boundary conditions for the velocity is considered and where the normal component of the magnetic field vanishes at the wall. Different flow regimes are obtained by starting from random initial velocity and magnetic fields with varying integral quantities. These regimes, equivalent to the ones observed by Ting, Matthaeus, and Montgomery [Phys. Fluids 29, 3261 (1986)] in periodic domains, are found to subsist in confined domains. The effect of solid boundaries on the energy decay and alignment properties is examined. The final states are characterized by functional relationships between velocity and magnetic field.