This paper presents a novel frequency-agile filtering power divider with constant absolute bandwidth and high selectivity. The proposed design utilizes four coupled resonators to obtain the two functions of power division and filtering. Synthesis of the filtering power divider is carried out and the design guidelines are given. Varactors are then loaded to the resonators to enable frequency tuning. By using capacitors as well as magnetic and electric coupling to obtain required external quality factors and coupling coefficients, a constant absolute bandwidth is maintained when tuning the frequency. To demonstrate the validity of the proposed design, a circuit was fabricated. Experimental results show that the operating frequency can be changed from 0.62 to 0.85 GHz with the 3-dB absolute bandwidth of 60 $\pm$ 2.5 MHz. Moreover, transmission zeros are generated near the passband, resulting in high selectivity for each tuning state. Comparisons between the measured and simulated results are presented and good agreement is observed.