\textit{V. de Smedt} in [Lett. Math. Phys. 31, No. 3, 225-231 (1994; Zbl 0797.16045)] proved that every finite-dimensional non-abelian Lie algebra over an algebraically closed field of characteristic \(0\) admits a nontrivial structure of a quasitriangular Lie bialgebra. The article under review is devoted to solving an analogous problem for Jordan algebras. By analogy with Lie bialgebras, the author defines the concepts of a triangular and quasitriangular Jordan bialgebras, using an analog of the Kantor-Köcher-Tits (KKT) construction for Jordan coalgebras. In the previous article [Algebra Logic 36, No. 1, 1-15 (1997; Zbl 0935.17014)], the author proved that if a Lie algebra \(L(J)\) obtained from a Jordan algebra \(J\) via the KKT construction admits a structure of a Lie bialgebra, then, under some natural restrictions, the algebra \(J\) admits a structure of a Jordan bialgebra. In the article under review, the author proves that a finite-dimensional Jordan algebra \(J\) over an algebraically closed field \(\Phi\) admits a nontrivial structure of a quasitriangular Jordan bialgebra provided that \(J\) is not a direct sum of fields, the algebras \(H(\Phi_2)\) and \(H(\Phi_3)\), null extensions of the field \(\Phi\), and of algebras with zero multiplication. The author also studies the cases in which a Jordan algebra fails to admit a nontrivial structure of a Jordan bialgebra and finds some sufficient conditions for a Jordan algebra to admit a nontrivial structure of a triangular Jordan bialgebra.