In the present work, the water transport dynamics and properties within CNT membranes are studied under the effect of pressure and electric field, using non-equilibrium molecular dynamics (NEMD) simulations. To reduce the pressure loss at the entrance and exit regions, the pore mouth shape has been changed to an hourglass shape for CNT membranes of (6,6), (8,8) and (12,12), inspired by aquaporins, which are capable of high water permeation. It is shown that as the pressure difference increases, the water flux enhancement due to change of pore mouth from plate to hourglass increases rapidly. Furthermore, it is found that the hourglass-shaped geometry has the highest impact on the reduction of entrance resistance within the pores with a smaller diameter. By applying an axial electric field at the entrance of the hourglass-shaped CNT membrane, the enhancement in water flux could be further improved. Interestingly, our results show that applying an electric field at the membrane entrance can cause two opposing effects on the water molecules transport: reduction of water molecules steric crowding at the entrance and increasing the average hydrogen bond numbers of the water molecules entering the tube. It seems that the former effect is dominant for low-intensity electric fields.