We study the propagation and collision of the compacton-like kinks in the system of an anharmoniclattice with a double well on -site potential by a direc algebraic method and numerical experiments. It is found that the localiazation of the compacton-like kinks is related to the nonlinear coupling parameter Cnl and the potetial barrier height v0 of the double well potential. The velocity of the propagation of the compacton like kinks is determined by the linear coupling parameter cl, the nlonlinear coupling parameter Cnl and the localization paratmeter q. Numerical experiments demonstrate that appropriate Cl is not detrimental to a stable propagation of the compacton-like kinks. However, the collision of compacton-like kiks and anti-kinds in the lattice with comparatively smallCl leads to the emergence of a discrete stationary breather and small amplitude nonlinear oscillation bacdground ,while moderate Cl results in the emergence of two deformed kinds with raditing oscillations and lower propagation velocities.