The work function of palladium (Pd) is known to be sensitive to hydrogen (H-2) via the formation of a surface dipole layer or Pd hydride. One approach to detect such a change in the work function is based on the formation of a Schottky barrier between Pd and a semiconductor. Here, we demonstrate a H-2 sensor operable at room temperature by assembling solution-processed, pre-separated semiconducting single-walled carbon nanotube (SWNT) network bridged by Pd source/drain (S/D) electrodes in a configuration of field-effect transistors (FETs) with a local back-gate electrode. To begin with, we observed that the H-2 response of the fabricated SWNT FETs can be enhanced in the linear operating regime, where the change in the work function of the Pd S/D electrodes by H2 can be effectively detected. We also explore the H-2 responses in various SWNT FETs with different physical dimensions to optimize the sensing performance.