In this work, we examine the vulnerability of language models to universal adversarial triggers (UATs). We propose a new white-box approach to the construction of layerwise UATs (LUATs), which searches the triggers by perturbing hidden layers of a network. On the example of three transformer models and three datasets from the GLUE benchmark, we demonstrate that our method provides better transferability in a model-to-model setting with an average gain of 9.3\% in the fooling rate over the baseline. Moreover, we investigate triggers transferability in the task-to-task setting. Using small subsets from the datasets similar to the target tasks for choosing a perturbed layer, we show that LUATs are more efficient than vanilla UATs by 7.1\% in the fooling rate.