In his previous article [|Algebra Logic 36, No. 1, 1-15 (1997); translation from Algebra Logika 36, No. 1, 3-25 (1997)], the author proved that if the Lie algebra \(L(J)\) built from a Jordan algebra \(J\) by the Kantor-Köcher-Tits construction admits the structure of a Lie bialgebra then, under certain natural restrictions, \(J\) admits the structure of a Jordan bialgebra. The converse problem is more complicated. The latter was solved by the author for Jordan algebras of type \(A^{(+)}\). In the article under review, this problem is solved for the Jordan algebras of symmetric elements. Let \(\Phi\) be a field and let \(A\) be an associative (Jordan) \(\Phi\)-algebra with coproduct \(\Delta\). Necessary and sufficient conditions are given for the pair \((A,\Delta)\) to be an associative (Jordan) bialgebra in the sense of Drinfel'd (\(D\)-bialgebra). Examples of associative \(D\)-bialgebras are presented. It is proven that an associative noncommutative finite-dimensional algebra over an algebraically closed field admits a nontrivial structure of an associative \(D\)-bialgebra with coproduct cocommutative on the center. The article is well written.