Purpose -The purpose of this paper is to aim at extending the method of exponential basis functions (EBF) to solve a class of problems with singularities. Design/methodology/approach -In the procedure of EBF a summation of EBF satisfying the governing differential equation with unknown constant coefficients is considered for the solution. These coefficients are determined by the satisfaction of prescribed boundary conditions through a collocation approach. The applied basis functions are available in the case of linear partial differential equations (PDEs) with constant coefficients. Moreover, the method contributes to yield highly accurate results with ultra convergence rates for problems with smooth solution. This leads EBF to offer many advantages for a variety of engineering problems. However, owing to the global and smooth nature of the bases, the performance of EBF deteriorates in problems with singularities. In the present study, some exponential-like influence functions are developed, and a few of them are added to original bases. Findings -The new bases are capable of forming the constitutive terms of the asymptotic solution near the singularity points and alleviate the aforementioned limitation. The appealing feature of this method is that all the advantages of EBF such as its simplicity and efficiency are completely preserved. Research limitations/implications -In its current form, EBF can only solve PDEs with constant coefficients. Originality/value -Application of the method to some benchmark problems demonstrates its robustness over some other boundary approximation methods. This research may pave the road for future investigations corresponding to a wide range of practical engineering problems.