The ionic current rectification (ICR) is a non-linear current-voltage response upon switching the polarity of the potential across nanopore, similar to the I-V response in the semiconductor diode. The ICR phenomenon finds several potential applications in micro/nano-fluidics (e.g., Bio-sensors and Lab-on-Chip applications). From a biological application viewpoint, most biological fluids (e.g., blood, saliva, mucus, etc.) exhibit non-Newtonian visco-elastic behavior; their rheological properties differ from Newtonian fluids. Therefore, the resultant flow-field should show an additional dependence on the rheological material properties of viscoelastic fluids such as fluid relaxation time (λ) and fluid extensibility (ε). Despite numerous potential applications, the comprehensive investigation of the viscoelastic behavior of the fluid on ionic concentration profile and ICR phenomena has not been attempted. ICR phenomena occur when the length scale and Debye layer thickness approaches of the same order. Therefore, this work extensively investigates the effect of viscoelasticity on the flow and ionic mass transfer along with the ICR phenomena in a single conical nanopore. The Poisson-Nernst-Planck (P-N-P) model coupled with momentum equations have been solved, for a wide range of conditions Deborah number, 1 ≤ De ≤ 100, Debye length parameter, 1 ≤ κRt ≤ 50, fluid extensibility parameter, 0.05 ≤ ε ≤ 0.25, applied electric potential, −40 ≤V ≤ 40, and surface charge density σ = −10 and −50. Four distinct novel characteristics of electro-osmotic flow (EOF) in a conical nanopore have been investigated here, namely (1) detailed structure of flow field and velocity distribution in viscoelastic fluids (2) influence of Deborah number and fluid extensibility parameter on ionic current rectification (ICR) (3) volumetric flow rate calculation as a function of Deborah number and fluid extensibility parameter (4) effect of viscoelastic parameters on concentration distribution of ions in the nanopore. At high applied voltage, both the extensibility parameter and Deborah number facilitate the ICR phenomena. In addition, the ICR phenomena are observed to be more pronounced at low values of κRt than the high values of κRt . This effect is due to the overlapping of the electric double layer at low values of κRt.