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The problem of the pursuit curve construction in the case when the tangent to pursuer’s motion trajectory passes at any time through the point representing the pursued is considered. A new approach to construct the pursuit curves using difference schemes is proposed. The proposed technique eliminates the need to derive the differential equations for the description of the pursuit curves, which is quite difficult task in the general case. In addition, the application of difference methods is justified in a situation where it is complicated to find the analytical solution of an existing differential equation and it is possible to obtain the pursuit curve only numerically. Various modifications of difference schemes respectively equivalent to the Euler, to the Adams – Bashforth and to the Milne methods are constructed. Their software implementation is realized by using the mathematical package Mathcad. We consider the case of a uniform rectilinear motion of the pursued whose differential equation describing the path of the pursuer and its analytical solution are known. We compare the numerical solutions obtained by the different methods with the well-known analytical solution. The error of the obtained numerical solutions is examined. Moreover, an application is considered illustrating the construction of the difference schemes for the case of an arbitrary trajectory of the pursued. Also, we extend the proposed method to the case of cyclic pursuit with several participants in the three-dimensional space. In particular, we construct a difference scheme equivalent to the Euler method for a three-dimensional analogue of the "bugs problem". The results obtained are demonstrated by means of animated examples for either two-dimensional or three-dimensional cases.