Cosmic shear has been identified as the method with the most potential to constrain dark energy. To capitalize on this potential, it is necessary to measure galaxy shapes with great accuracy, which in turn requires a detailed model for the image blurring by the telescope and atmosphere, the point spread function (PSF). In general, the PSF varies with wavelength and therefore the PSF integrated over an observing filter depends on the spectrum of the object. For a typical galaxy the spectrum varies across the galaxy image, thus the PSF depends on the position within the image. We estimate the bias on the shear due to such colour gradients by modelling galaxies using two co-centred, co-elliptical Sérsic profiles, each with a different spectrum. We estimate the effect of ignoring colour gradients and find the shear bias from a single galaxy can be very large depending on the properties of the galaxy. We find that halving the filter width reduces the shear bias by a factor of about 5. We show that, to the first order, tomographic cosmic shear two point statistics depend on the mean shear bias over the galaxy population at a given redshift. For a single broad filter, and averaging over a small galaxy catalogue from Simard et al., we find a mean shear bias which is subdominant to the predicted statistical errors for future cosmic shear surveys. However, the true mean shear bias may exceed the statistical errors, depending on how accurately the catalogue represents the observed distribution of galaxies in the cosmic shear survey. We then investigate the bias on the shear for two-filter imaging and find that the bias is reduced by at least an order of magnitude. Lastly, we find that it is possible to calibrate galaxies for which colour gradients were ignored using two-filter imaging of a fair sample of noisy galaxies, if the galaxy model is known. For a signal-to-noise ratio of 25 the number of galaxies required in each tomographic redshift bin is of the order of 104.