Deep fully connected networks are often considered 'universal approximators' that are capable of learning any function. In this article, we utilize this particular property of deep neural networks (DNNs) to estimate normalized cross correlation as a function of spatial lag (i.e., spatial coherence functions) for applications in coherence-based beamforming, specifically short-lag spatial coherence (SLSC) beamforming. We detail the composition, assess the performance, and evaluate the computational efficiency of CohereNet, our custom fully connected DNN, which was trained to estimate the spatial coherence functions of in vivo breast data from 18 unique patients. CohereNet performance was evaluated on in vivo breast data from three additional patients who were not included during training, as well as data from in vivo liver and tissue mimicking phantoms scanned with a variety of ultrasound transducer array geometries and two different ultrasound systems. The mean correlation between the SLSC images computed on a central processing unit (CPU) and the corresponding DNN SLSC images created with CohereNet was 0.93 across the entire test set. The DNN SLSC approach was up to 3.4 times faster than the CPU SLSC approach, with similar computational speed, less variability in computational times, and improved image quality compared with a graphical processing unit (GPU)-based SLSC approach. These results are promising for the application of deep learning to estimate correlation functions derived from ultrasound data in multiple areas of ultrasound imaging and beamforming (e.g., speckle tracking, elastography, and blood flow estimation), possibly replacing GPU-based approaches in low-power, remote, and synchronization-dependent applications.