By means of numerical simulation the elastohydrodynamic lubricating film formation during start-up of motion of a line contact from rest is studied. The transition from solid contact to lubricated contact is of importance in investigating the start-up process and its effects on bearing performance. The numerical results of and others, concerning point contact and Newtonian lubricant, are not in good agreement with the known experimental data. The numerical film thickness is significantly smaller than the experimental values and the experimental velocity at which lubricant travels through the conjunction does not equal the entrainment velocity of numerical results. In the current paper, a lubricant with viscoelastic properties is considered. The Maxwell model of a lubricant at pure rolling of elastic cylinders is used. Also, as well as in work of Usov (2008) the model assumes the division of the whole area of contact into three zones: the dry contact of surfaces, the transient zone and the zone in which the surfaces are separated by a liquid lubricant layer. The rigid body model is assumed for the lubricant in the transient zone. The effects of viscoelasticity and size of transient zone on film thickness and the velocity at which the lubricant moves through the contact are studied. It is shown that viscoelasticity raises film thickness in a start of the motion and lowers the velocity at which the lubricant moves through the contact. This velocity is lower than the entrainment velocity in the first half of the contact, as in the experimental results with point contact. In the second half of the contact, this velocity may be lower or higher than the entrainment velocity depending on the size of the transient zone. The film thickness may increase considerably at the start of motion when the size of the transient zone increases.