In the unbalance response analysis of rotors supported by squeeze-film dampers with centering springs, the fluid-film forces are usually computed based on the assumption that the rotor exhibits a circular centered whirl orbit motion. The validity of this assumption is, however, limited to the ideal case of squeeze-film dampers with centering springs that are perfectly adjusted to offset the gravitational force. In most practical applications, eccentric operation of these dampers is almost unavoidable since precise setting of the centering springs in a real environment is usually not possible. In this paper the bifurcations in the response of a rigid rotor in eccentric squeeze-film dampers are investigated. The values of the bearing parameter (B), gravity parameter (W) and spring parameter (S) are respectively fixed at 0.015, 0.05 and 0.3, while the unbalance parameter (U) is varied from 0.05 to 0.8. The results indicated that the rotor might lose its stability due to period-doubling and saddle node bifurcations. Chaotic response of the rotor was also observed for 0.365 < U < 0.367 and 0.381 < U < 0.392. The transitions to chaos in these two regimes were respectively via the period-doubling and type 3 intermittency routes. The levels of rotor unbalance where non-synchronous and chaotic motions were observed in this study are only an order of magnitude higher than the specified levels for rigid rotors. Such levels of unbalance may easily occur in practice due to in-service erosion or in the event of a partial or an entire blade loss.