Objective: Transcranial direct current stimulation (tDCS) is a non-invasive brain stimulation technique used to generate conduction currents in the head and disrupt brain functions. To rapidly evaluate the tDCS-induced current density in near real-time, this paper proposes a deep learning-based emulator, named DeeptDCS. Methods: The emulator leverages Attention U-net taking the volume conductor models (VCMs) of head tissues as inputs and outputting the three-dimensional current density distribution across the entire head. The electrode configurations are also incorporated into VCMs without increasing the number of input channels; this enables the straightforward incorporation of the non-parametric features of electrodes (e.g., thickness, shape, size, and position) in the training and testing of the proposed emulator. Results: Attention U-net outperforms standard U-net and its other three variants (Residual U-net, Attention Residual U-net, and Multi-scale Residual U-net) in terms of accuracy. The generalization ability of DeeptDCS to non-trained electrode configurations can be greatly enhanced through fine-tuning the model. The computational time required by one emulation via DeeptDCS is a fraction of a second. Conclusion: DeeptDCS is at least two orders of magnitudes faster than a physics-based open-source simulator, while providing satisfactorily accurate results. Significance: The high computational efficiency permits the use of DeeptDCS in applications requiring its repetitive execution, such as uncertainty quantification and optimization studies of tDCS.