A system includes a turbomachine rotor having a shaft and turbomachine blades coupled to the shaft. The system also includes a turbomachine stator having a shroud surrounding the turbomachine blades of the turbomachine rotor. Further, the system includes a cooling channel having at least a first portion of the cooling channel extending upstream of a final stage of a compressor of the system, where the cooling channel is configured to receive cooled compressed air from the compressor and direct the cooled compressed air adjacent to the turbomachine rotor to reduce thermal expansion and/or axial displacement of the turbomachine rotor.