We obtained high-resolution spectra of the ultracool M-dwarf TRAPPIST-1 during the transit of its planet "b" using two high-dispersion near-infrared spectrographs, the Infrared Doppler (IRD) instrument on the Subaru 8.2m telescope, and the Habitable Zone Planet Finder (HPF) instrument on the 10 m Hobby-Eberly Telescope. These spectroscopic observations are complemented by a photometric transit observation for planet "b" using the APO/ARCTIC, which assisted us in capturing the correct transit times for our transit spectroscopy. Using the data obtained by the new IRD and HPF observations, as well as the prior transit observations of planets "b," "e" and "f" from IRD, we attempt to constrain the atmospheric escape of the planet using the He I triplet 10830 Å absorption line. We do not detect evidence for any primordial extended H-He atmospheres in all three planets. To limit any planet-related absorption, we place an upper limit on the equivalent widths of <7.754 mÅ for planet "b," <10.458 mÅ for planet "e," <4.143 mÅ for planet "f" at 95% confidence from the IRD data, and <3.467 mÅ for planet "b" at 95% confidence from HPF data. Using these limits along with a solar-like composition isothermal Parker wind model, we attempt to constrain the mass-loss rates for the three planets. For TRAPPIST-1b, our models exclude the highest possible energy-limited rate for a wind temperature <5000 K. This nondetection of extended atmospheres with low mean-molecular weights in all three planets aids in further constraining their atmospheric composition by steering the focus toward the search of high-molecular-weight species in their atmospheres. *Based on data collected at Subaru Telescope, operated by the National Astronomical Observatory of Japan, Hobby-Eberly Telescope operated by The University of Texas McDonald Observatory, and ARC 3.5m Telescope at Apache Point Observatory.