The galloping motion of an elastic circular cylinder, with oscillations restricted to a plane normal to the incident uniform flow, when it is located in the close neighbourhood of a plane boundary is investigated theoretically by using a quasi-steady assumption; that the cylinder and the flow are both parallel to the boundary. Closed form expressions are presented to describe the cylinder behaviour: namely, the variation of the reduced (relative) amplitude with the reduced velocity, the relative mean position of the vibration with the dimensionless free stream dynamic pressure and the relative motion frequency with the reduced amplitude. The first variation and also that of the cylinder mean position with the flow velocity approach asymptotically constant values, the second one is a function only of the lift coefficients, while the last one depends on both the dimensionless free stream dynamic pressure and the lift coefficients. The mathematical model predicts that an increase in the cylinder mass or structural damping causes the galloping to occur at a higher flow velocity. Simple expressions are presented for the prediction of the reduced velocity and the cylinder motion frequency at the galloping onset. The theory indicates that the galloping instability occurs only when the gap between the cylinder and the boundary is very small.