Bilevel optimization problems have been actively studied in recent machine learning research due to their broad applications. In this work, we investigate single-loop methods with iterative differentiation (ITD) for nonconvex bilevel optimization problems. For deterministic bilevel problems, we propose an efficient single-loop ITD-type method (ES-ITDM). Our method employs historical updates to approximate the hypergradient. More importantly, based on ES-ITDM, we propose a new method that avoids computing Hessians. This Hessian-free method requires fewer backpropagations and thus has a lower computational cost. We analyze the convergence properties of the proposed methods in two aspects. We provide the convergence rates of the sequences generated by ES-ITD based on the Kurdyka-\L ojasiewicz (KL) property. We also show that the Hessian-free stochastic ES-ITDM has the best-known complexity while has cheaper computation. The empirical studies show that our Hessian-free stochastic variant is more efficient than existing Hessian-free methods and other state-of-the-art bilevel optimization approaches.