The solvability of the rational Hermite-interpolation problem is investigated through an approach similar to that developed in an earlier paper [1] for the ordinary case. However, the subsequent deduction of analogous results involves much complications. The Quasi-Rational Hermite Interpolant $r_{mn}^{*}$ is introduced. In the case of $r_{mn}^{*}$ being nondegenerate, its explicit expression is given. Working with the notion of l-fold unattainable point and using algebraic elaboration, we have successively established several theorems concerning interpolating properties of $r_{mn}^{*}$ and, in particular, obtained existence theorems for the solution of the proposed problem.