The global stability analysis is presented for the boundary layer on a thin, long circular cylinder aligned with a free-stream under the effect of suction and injection. A steady, two-dimensional laminar base flow solution is obtained by OpenFOAM software. The two-dimensional temporal stability equations are obtained by standard procedure from the equation of motions for incompressible flow in cylindrical coordinates. The spectral method is employed for spatial discretization of the stability equations. The eigenvalue problem (EVP) is established by formulating the stability equations along with appropriate boundary conditions in both the streamwise (x) and radial (r) directions. The EVP is solved using the Arnoldi iteration method, specifically implemented through the ARPACK package. The findings of an axisymmetric boundary layer (ABL) and a flat-plate boundary layer (FBL) are compared and found that transverse curvature strongly stabilizes the cylinder boundary layer. Also, ABL is found modally stable due to suction, while the boundary layer instability occurs because of injection even though Reynolds number (195) and intensity (0.5% of \(U_{\infty }\) ) are very small. The 2D spatial framework of temporal modes shows the wave packets are suppressed due to the suction and amplified due to the injection.