The paper deals with the problem of quasistatic frictionless contact between an elastic body and a foundation. The elasticity operator is assumed to vanish for zero strain, to be Lipschitz continuous and strictly monotone with respect to the strain as well as Lebesgue measurable on the domain occupied by the body. The contact is modelled by normal compliance in such a way that the penetration is limited and restricted to unilateral contraints. In this problem we take into account adhesion which is modelled by a surface variable, the bonding field, whose evolution is described by a first-order differential equation. We derive a variational formulation of the mechanical problem and we establish an existence and uniqueness result by using arguments of time-dependent variational inequalities, differential equations and the Banach fixed-point theorem. Moreover, using compactness properties we study a regularized problem which has a unique solution and we obtain the solution of the original model by passing to the limit as the regularization parameter converges to zero.