We propose theoretically and numerically, for the first time, the generation of novel partially coherent truncated Airy beams (NPCTABs) with Airy-like distributions for both intensity and degree of coherence via Fourier phase processing. We demonstrate a clear link between the magnitude and frequency of intensity and degree of coherence distributions oscillations of generated beams, and the source coherence and the phase screen parameter. Thus, the source coherence and phase can serve as convenient parameters to control the intensity and degree of the coherence of NPCTABs. Furthermore, we discover that NPCTABs are more stable than the fully coherent truncated Airy beams (FCTABs) during their propagation in free space and can maintain their Airy-like profile for an extended propagation distance. The interesting and tunable characteristics of these novel beams may find applications in particle trapping, phase retrieval, and optical imaging.