We present a novel pipeline that uses a convolutional neural network (CNN) to improve the detection capability of near-Earth asteroids (NEAs) in the context of planetary defense. Our work aims to minimize the dependency on human intervention of the current approach adopted by the Zwicky Transient Facility (ZTF). The target NEAs have a high proper motion of up to tens of degrees per day and thus appear as streaks of light in the images. We trained our CNNs to detect these streaks using three datasets: a set with real asteroid streaks, a set with synthetic (i.e., simulated) streaks and a mixed set, and tested the resultant models on real survey images. The results achieved were almost identical across the three models: 0.843 +/- 0.005 in completeness and 0.820 +/- 0.025 in precision. The bias on streak measurements reported by the CNNs was 1.84 +/- 0.03 pixels in streak position, (0.817 +/- 0.026)degrees in streak angle and -0.048 +/- 0.003 in fractional bias in streak length (computed as the absolute length bias over the streak length, with the negative sign indicating an underestimation). We compared the performance of our CNN trained with a mix of synthetic and real streaks to that of the ZTF human scanners by analyzing a set of 317 streaks flagged as valid by the scanners. Our pipeline detected 80% of the streaks found by the scanners and 697 additional streaks that were subsequently verified by the scanners to be valid streaks. These results suggest that our automated pipeline can complement the work of the human scanners at no cost for the precision and find more objects than the current approach. They also prove that the synthetic streaks were realistic enough to be used for augmenting training sets when insufficient real streaks are available or exploring the simulation of streaks with unusual characteristics that have not yet been detected.