Strong near-shore earthquakes are the most frequent sources of tsunamis in many oceans of the world. In the framework of the nonlinear shallow-water theory, the initial sea-surface tsunami elevation is assumed to equal the sea-floor co-seismic displacement produced by the seismic event. This is quantified by means of the analytical formulas due to Okada (1985, 1992), dealing with seismic faults buried in an elastic medium. In this work the propagation of tsunamis is studied along two-dimensional profiles on an idealized constant-slope sea bed, an approximation that allows one to reduce the governing nonlinear equations to a linear problem by means of the classical Carrier & Greenspan (1958) approach. We introduce an analytical solution that is sufficiently general to account for initial conditions associated with paradigmatic cases of sea-bottom deformations produced by near-shore earthquakes, such as subsidence or uplift of the coastal area, and can be also used to treat more complex deformations. The main result is that the amplification of the tsunami height at the coast is found to range between approximately 1 and 2. The amplification is around 1 for tsunamis induced by earthquakes with their epicentre inland and tends to grow as the fault moves seaward. We restrict our analysis to earthquakes that dislocate the shore region. Within the class of sources that we consider, the tsunamis that are most amplified are the ones having initial profiles with a crest–trough–crest system or conversely with a trough–crest–trough system. The bottom slope is found to have no effect on tsunami run-ups and run-downs, but to influence tsunami periods and tsunami speed remarkably. Breaking analysis shows that wave breaking does not occur if the initial wave height is less than 8–9 m, and that the simplest sea-level profiles, which are associated with earthquakes with their epicentre on land, are not expected to break even if their initial height exceeds 19 m.