Integrated circuits are provided with circuitry such as multiplexers that can be selectively configured to route different power supply voltages to different circuit blocks on the integrated circuits. The circuit blocks may contain memory elements that are powered by the power supply voltages and that provide corresponding static output control signals at magnitudes that are determined by the power supply voltages. The control signals from the memory elements may be applied to the gates of transistors in the circuit blocks. Logic on an integrated circuit may be powered at a given power supply voltage level. The memory elements may provide their output signals at overdrive voltage levels that are elevated with respect to the given power supply voltage level during high speed operation and may provide their output signals at relatively lower voltage levels that are less than the overdrive voltage during low power operation.