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The Event Horizon Telescope (EHT) collaboration recently obtained the first images of the surroundings of the supermassive compact object M87* at the center of the galaxy M87. This provides a fascinating probe of the properties of matter and radiation in strong gravitational fields. It is important to determine from the analysis of these results what can and cannot be inferred about the nature of spacetime around M87*Aims. We want to develop a simple analytic disk model for the accretion flow of M87*. Compared to general-relativistic magnetohydrodynamic models, this new approach has the advantage that it is independent of the turbulent character of the flow and is controlled by only a few easy-to-interpret, physically meaningful parameters. We want to use this model to predict the image of M87*, assuming that it is either a Kerr black hole or an alternative compact object.Methods. We computed the synchrotron emission from the disk model and propagate the resulting light rays to the far-away observer by means of relativistic ray tracing. Such computations were performed assuming different spacetimes, such as Kerr, Minkowski, nonrotating ultracompact star, rotating boson star, or Lamy spinning wormhole. We performed numerical fits of these models to the EHT data.Results. We discuss the highly lensed features of Kerr images and show that they are intrinsically linked to the accretion-flow properties and not only to gravitation. This fact is illustrated by the notion of the secondary ring, which we introduce. Our model of a spinning Kerr black hole predicts mass and orientation consistent with the EHT interpretation. The non-Kerr images result in a similar quality of numerical fits and may appear very similar to Kerr images, once blurred to the EHT resolution. This implies that a strong test of the Kerr spacetime may be out of reach with the current data. We note that future developments of the EHT could alter this situation.Conclusions. Our results show the importance of studying alternatives to the Kerr spacetime to be able to test the Kerr paradigm unambiguously. More sophisticated treatments of non-Kerr spacetimes and more advanced observations are needed to proceed further in this direction.