Few-cycle infrared laser pulses are of utmost importance for a wide range of scientific application in quantum optics, ultrafast spectroscopy and strong-field physics. Optical parametric chirped-pulse amplification (OPCPA), owing to its unique properties including ultra-broad gain bandwidth, large gain efficiency and high wavelength tunability, has attracted lots of attention and become the most widely used source for such types of laser pulses. In a conventional OPCPA system, a long-duration (picosecond or nanosecond), high-energy pulse is usually selected as the pump, and the broadband signal pulse is temporally stretched to match the pump duration for an efficient energy conversion. While in this case, phase-matching can only be realized for a particular signal wavelength. As the instantaneous wavelength of the signal shift away from the phase-matched components, the phase-mismatching increases and the gain efficiency reduces, resulting in the narrowing of the gain bandwidth and limiting the available duration of the output pulses. A dual-pump optical parametric chirped-pulse amplification scheme, which can effectively suppress the chirp-induced bandwidth-narrowing effect, is theoretically investigated for generating broadband IR pulses at degeneracy. The initial signal pulse is linearly chirped, two broadband pump pulses are linearly chirped with inverse GDDs (Group Delay Dispersion) to fit the phase-matching curve. In this case, phase-matching can be realized through a wide spectral range. As the simulation shows, the output spectrum spans from 1.3 µm to 2.1 µm, supporting a transform-limited (TL) pulse with a duration of 9.0 fs. The output pulse energy is 3.97 mJ, corresponding to a conversion efficiency of 22.6%. The millijoule pulses can be compressed to a near-TL duration of 10.1 fs by simply removing the linear chirp with a pair of gratings or prisms. Furthermore, both the pump and signal pulses used in this scheme are produced with only one femtosecond Ti:sapphire laser, which remarkably simplifies the synchronization between the interacting pulses. The versatile dual-pump OPCPA scheme with such simplicity and energy-scalability will contribute to ultrafast community and be particularly useful as a driving or controlling field for the generation of ultrafast coherent x-ray supercontinuum. This work was supported by the NNSF of China under grants 11204095, 11234004 and 61275126, and the 973 Program of China under grant 2011CB808103.