One of the major challenges for the radio detection of extensive air showers, as encountered by the Giant Radio Array for Neutrino Detection (GRAND), is the requirement of an autonomous radio self-trigger. This work presents the current development of self-triggering techniques at the detection-unit level---the so-called first-level trigger (FLT)---in the context of the NUTRIG project. A second-level trigger (SLT) at the array level is described in a separate contribution. Two FLT methods are described, based on a template-fitting algorithm and a convolutional neural network (CNN). In this work, we compare the preliminary offline performance of both FLT methods in terms of signal selection efficiency and background rejection efficiency. We find that for both methods, ${\gtrsim}40\%$ of the background can be rejected if a signal selection efficiency of 90\% is required at the $5\sigma$ level.