Under the epidemic of the novel coronavirus disease 2019 (COVID-19), chest X-ray computed tomography imaging is being used for effectively screening COVID-19 patients. The development of computer-aided systems based on deep neural networks (DNNs) has been advanced, to rapidly and accurately detect COVID-19 cases, because the need for expert radiologists, who are limited in number, forms a bottleneck for the screening. However, so far, the vulnerability of DNN-based systems has been poorly evaluated, although DNNs are vulnerable to a single perturbation, called universal adversarial perturbation (UAP), which can induce DNN failure in most classification tasks. Thus, we focus on representative DNN models for detecting COVID-19 cases from chest X-ray images and evaluate their vulnerability to UAPs generated using simple iterative algorithms. We consider nontargeted UAPs, which cause a task failure resulting in an input being assigned an incorrect label, and targeted UAPs, which cause the DNN to classify an input into a specific class. The results demonstrate that the models are vulnerable to nontargeted and targeted UAPs, even in case of small UAPs. In particular, 2% norm of the UPAs to the average norm of an image in the image dataset achieves >85% and >90% success rates for the nontargeted and targeted attacks, respectively. Due to the nontargeted UAPs, the DNN models judge most chest X-ray images as COVID-19 cases. The targeted UAPs make the DNN models classify most chest X-ray images into a given target class. The results indicate that careful consideration is required in practical applications of DNNs to COVID-19 diagnosis; in particular, they emphasize the need for strategies to address security concerns. As an example, we show that iterative fine-tuning of the DNN models using UAPs improves the robustness of the DNN models against UAPs.