Traditionally, the output noise and power supply rejection of low-dropout regulators (LDOs) are optimized to minimize power supply fluctuations, reducing their impact on the low-frequency noise of target voltage-controlled oscillators (VCOs). However, this sequential design approach does not fully address the trade-offs between high-frequency and LDO-induced low-frequency phase noise. To overcome this limitation, this paper presents a co-design method for low phase-noise LC-tank VCOs powered by LDOs. It is difficult to carry out the co-design using traditional manual design techniques. Hence, an efficient AI-driven EDA algorithm is used. To validate the proposed method, a 5.6 GHz LC-tank VCO with an integrated LDO is designed using a 65 nm CMOS process. Simulations show that the co-design method improves phase noise by 1.2 dB at a 1 MHz offset and reduces dynamic power consumption by 28.8%, with FoM increased by 2.4 dBc/Hz compared to the conventional sequential design method.