In air-cooled data centers, cooling systems account for nearly half of the total energy consumption to maintain a safe and reliable thermal environment for IT equipment. Optimizing cooling system operation and workload allocation is crucial for improving thermal management and reducing cooling energy consumption. In this study, a novel chip-temperature-aware model predictive control (MPC) strategy is proposed to jointly optimize server workload allocation and computer room air conditioner (CRAC) operating parameters. First, the thermal dynamics and power consumption models of the data center were developed to characterize the interdependence between server computational workloads and thermal states, spanning from chip-level to room-level. Subsequently, two chip-temperature-aware MPC controllers (energy-prioritized and workload-prioritized) were designed to minimize CRAC cooling power while maintaining safe server thermal conditions. Finally, the performance and key design factors of the proposed controllers were evaluated using a data center simulation model. The results demonstrate that the chip-temperature-aware MPC strategies, compared with an inlet-temperature-aware MPC baseline, significantly reduce cooling energy consumption while maintaining server thermal stability, highlighting their potential for energy-efficient operation in comparable air-cooled data centers.