We study the steady three-dimensional flow field and bed topography in a channel with sinusoidally varying width, under the assumptions of small-amplitude width variations and sufficiently wide channel to neglect nonlinear effects and sidewall effects. The aim of the work is to investigate the role of width variations in producing channel bifurcation in braided rivers. We infer incipient bifurcation in cases where the growth of a central bar leads to planimetric instability of the channel, i.e. when the given infinitesimal width perturbation is enhanced. Results of the three-dimensional model suggest that the equilibrium bottom profile mainly consists of a purely longitudinal component, uniformly distributed over the cross-section, which induces deposition at the wide section and scour at the constriction, and of a transverse component in the form of a central bar (wide sections) and scour (constrictions), with longitudinal wavelength equal to that of width variations. A comparison between the results of the three-dimensional model and those obtained by means of a two-dimensional depth-averaged approach shows that the transverse component is mainly related to three-dimensional effects. Theoretical findings display a satisfactory agreement with results of flume experiments. Transverse variations are responsible for the planimetric instability of the channel; we find that in the range of values of Shields stress typical of braided rivers, the incipient bifurcation is enhanced as the width ratio of the channel increases.