Pulsatile flow of a conducting Jeffrey fluid through a porous elastic tube with variable cross-section is investigated in this paper. The Jeffrey fluid flow under the effect of transverse magnetic field is considered to understand the nature of conductive physiological fluids such as movement of blood in small arteries of heart muscle. The problem is formulated by considering the radial motion of the tube wall. The analytical expressions of the axial and transverse velocity for both steady and fluctuating parts are derived. The influence of different pertinent parameter on axial velocity distribution for different instants of times is analyzed through graphs. The variation in the velocity field for two different geometries is discussed. The results reveal that the effect of frequency parameter on fluctuating axial velocity is significant. The influence of magnetic field and Jeffrey parameter has pronounced effect on variation of fluctuating axial velocity. Also it is observed that the geometry has a remarkable effect on axial velocity distribution. The results obtained for the present oscillatory flow characteristics reveal many interesting behaviors that warrant further study of conductive physiological fluid flows in an elastic tube with variable cross-section. In the absence of Magnetic field and Jeffrey parameter the present results are similar to the observations of Misra and Ghosh [8]. © 2018 Elsevier Ltd