In thermofluid dynamics, free convection flows external to different geometries such as cylinders, ellipses, spheres, curved walls, wavy plates, and cones play a major role in various industrial and process engineering systems. The thermal buoyancy force associated with natural convection flows can exert a critical role in determining skin friction and heat transfer rates at the boundary. In thermal engineering, natural convection flows from cones has gained exceptional interest. A theoretical analysis is developed to investigate the nonlinear, steady-state, laminar, non-isothermal convection boundary layer flows of viscoelastic fluid from a vertical permeable cone with a power-law variation in both temperature and concentration. The Jeffery’s viscoelastic model simulated the non-Newtonian characteristics of polymers, which constitutes a novelty of the present work. The transformed conservation equation for linear momentum, energy, and concentration are solved numerically under physically viable boundary conditions using the finite-difference Keller box scheme. The impact of Deborah number (De), ratio of relaxation to retardation time (λ), surface suction/injection parameter (fw), power-law exponent (n), buoyancy ratio parameter (N), and dimensionless tangential coordinate (ξ) on velocity, surface temperature, concentration, local skin friction, heat transfer rate, and mass transfer rate in the boundary layer regime is presented graphically. It is observed that increasing values of De reduces velocity whereas the temperature and concentration are increased slightly. Increasing λ enhance velocity, however, reduces temperature and concentration slightly. The heat and mass transfer rate are found to decrease with increase in De and increase with increasing values of λ. The skin friction is found to decrease with a rise in De, whereas it is elevated with increasing values of λ. Increasing values of fw and n decelerates the flow and also cools the boundary layer, i.e., reduces temperature and also concentration. The study is relevant to chemical engineering systems, solvent, and polymeric processes. © 2018, The Brazilian Society of Mechanical Sciences and Engineering.