The dengue virus (DENV) nonstructural protein 5 (NS5) contains both an N-terminal methyltransferase domain and a C-terminal RNA-dependent RNA polymerase domain. Polymerase activity is responsible for viral RNA synthesis by a de novo initiation mechanism and represents an attractive target for antiviral therapy. The incidence of DENV has grown rapidly and it is now estimated that half of the human population is at risk of becoming infected with this virus. Despite this, there are no effective drugs to treat DENV infections. The present in silico study aimed at finding new inhibitors of the NS5 RNA-dependent RNA polymerase of the four serotypes of DENV. We used a chemical library comprising 372,792 nonnucleotide compounds (around 325,319 natural compounds) to perform molecular docking experiments against a binding site of the RNA template tunnel of the virus polymerase. Compounds with high negative free energy variation (ΔG <−10.5 kcal/mol) were selected as putative inhibitors. Additional filters for favorable druggability and good absorption, distribution, metabolism, excretion, and toxicity were applied. Finally, after the screening process was completed, we identified 39 compounds as lead DENV polymerase inhibitor candidates. Potentially, these compounds could act as efficient DENV polymerase inhibitors in vitro and in vivo.