Decision-makers must often balance the desire to accumulate information with the costs of protracted deliberation. Optimal, reward-maximizing decision-making can require dynamic adjustment of this speed/accuracy trade-off over the course of a single decision. However, it is unclear whether humans are capable of such time-dependent adjustments. Here, we identify several signatures of time-dependency in human perceptual decision-making and highlight their possible neural source. Behavioural and model-based analyses reveal that subjects respond to deadline-induced speed pressure by lowering their criterion on accumulated perceptual evidence as the deadline approaches. In the brain, this effect is reflected in evidence-independent urgency that pushes decision-related motor preparation signals closer to a fixed threshold. Moreover, we show that global modulation of neural gain, as indexed by task-related fluctuations in pupil diameter, is a plausible biophysical mechanism for the generation of this urgency. These findings establish context-sensitive time-dependency as a critical feature of human decision-making.