Quantitative susceptibility mapping (QSM) is a valuable MRI post-processing technique that quantifies the magnetic susceptibility of body tissue from phase data. However, the traditional QSM reconstruction pipeline involves multiple non-trivial steps, including phase unwrapping, background field removal, and dipole inversion. These intermediate steps not only increase the reconstruction time but amplify noise and errors. This study develops a large-stencil Laplacian preprocessed deep learning-based neural network for near instant quantitative field and susceptibility mapping (i.e., iQFM and iQSM) from raw MR phase data. The proposed iQFM and iQSM methods were compared with established reconstruction pipelines on simulated and in vivo datasets. In addition, experiments on patients with intracranial hemorrhage and multiple sclerosis were also performed to test the generalization of the novel neural networks. The proposed iQFM and iQSM methods yielded comparable results to multi-step methods in healthy subjects while dramatically improving reconstruction accuracies on intracranial hemorrhages with large susceptibilities. The reconstruction time was also substantially shortened from minutes using multi-step methods to only 30 milliseconds using the trained iQFM and iQSM neural networks.