Locally-active memristors (LAMs) have broad application prospects in neuromorphic computing due to their nanoscale, low-power consumption, and local activity. A LAM combined with a DC bias and a reactance element (capacitor or inductor) can form an oscillator. Since the LAM exhibits capacitive or inductive impedance characteristic at the specific operating points, it can replace the reactance element and further implement a reactance-less oscillator. Two different LAMs-based reactance-less oscillators are proposed. One is based on two voltage-controlled (VC) LAMs, and the other mainly contains two current-controlled (CC) LAMs. Local activity theory, Hopf bifurcation theory, and the small-signal analysis method are used to quantitatively deduce the oscillation condition of the proposed circuit. A simple VC LAM mathematical model is taken as an example to conduct the analysis and verification. Both simulation and experiment results confirm the feasibility and practicability of the proposed LAMs-based reactance-less oscillator.