In this paper, the authors consider a hyperbolic-parabolic singular perturbation for quasilinear equations of Kirchhoff type. The authors show estimates of the difference between the solution \(u_\varepsilon\) of a quasilinear hyperbolic equation and the solution \(v_\varepsilon\) of the corresponding parabolic equation depending on \(v_\varepsilon\), by regarding them as the solutions of the linear hyperbolic equations and the parabolic equations with same constant of \(A\). Next they show time decay estimates of the difference between \(v_\varepsilon\) and the solution \(w\) of the original parabolic equation. From these estimates, they obtain time decay estimates of the singular-perturbation problem for Kirchhoff equation.