The deformation of a two-dimensional inextensible elastic cell in an inviscid uniform stream with circulation is investigated. An asymptotic expansion based on a conformal mapping is used to obtain equilibria for low far-field flow speeds, and fully nonlinear solutions are obtained numerically. Expanding upon the results of Blyth and Părău (2013) and Yorkston et al. (2020) for an elastic cell in a uniform stream with zero circulation, it is shown that the nature of the cell deformation in response to circulation depends on whether the transmural pressure exceeds a series of critical values. Below the first of these critical values, the deformed cell is elongated vertically against the stream, and the circulation acts to reduce the deformation of the cell from the circular rest-state, while above this critical pressure the deformed cell elongates horizontally parallel to the flow, with stronger circulation resulting in more severe cell deformation until self-intersection. The solution branches which emerge at the second critical transmural pressure are found to form a closed loop in parameter space, which shrinks in size as the circulation is increased to a critical value at which the solution branch vanishes. We also present a set solution branches distinct from those found by Yorkston et al. (2020), which become dominant for large values of circulation
