A quasi-ideal Q of a ring A is an additive subgroup of A such that A\(Q\cap QA\subseteq Q\). The intersection of a left ideal and a right ideal is always a quasi-ideal but the converse is not true (a counterexample was constructed by A. H. Clifford [see \textit{O. Steinfeld}, Quasi-ideals in rings and semigroups (1978; Zbl 0403.16001), p. 8]). A quasi-ideal Q is said to have the intersection property if it is the intersection of a left and a right ideal. The main results of the present paper exhibit for various rings that they possess quasi-ideals which do not have the intersection property. Such is for instance the contracted semigroup ring \(R_ 0[S]\) where R is an arbitrary ring with identity and S is a semigroup with zero satisfying a condition which is too technical to be reproduced here. The author also constructs a skew polynomial ring without zero divisors in which not every quasi-ideal has the intersection property although every minimal quasi-ideal has it. Similar examples of contracted semigroup rings are also presented.