Abstract: Let S be a regular semigroup with an inverse transversal. Following the study of the lifting property of ele-ments in the quotient semigroup of S and the lifting property of Green's relations in the homomorphic image of S, congruences on S generated by Green's relations and by restriction of Green's relations to the important subsemi-groups of S are investigated. The classes of semigroups related to these congruences are described.