The study of creeping motion of viscoelastic fluid around a rotating rigid torus is investigated. The problem is solved within the frame of slow flow approximation. The equations of motion governing the first and second-order are formulated and solved for the first-order only in this paper. However, the solution of the second-order equations will be the subject of a part two of this series of papers. Analytically, Laplace's equation is solved via the usual method of separation of variables. This method shows that, the solution is given in a form of infinite sums over Legendre functions of the first and second kinds. From the obtained solution it is found that, the leading term of the velocity represents the Newtonian flow. The second-order term shows that, the only non vanishing term is the stream function, which describes a secondary flow domain. The distribution of the surface traction at the toroid surface is calculated and discussed. Considering hydrodynamically conditions, the effects of toroidal geometrical parameters on the flow field are investigated in detail.