Abstract Observations by the High-Altitude Water Cherenkov Observatory (HAWC) and Milagro have detected spatially extended TeV sources surrounding middle-aged (t ∼ 100–400 kyr) pulsars like Geminga and PSR B0656+14, which have been named “TeV halos,” representing very extended TeV pulsar wind nebulae (PWNe) powered by relatively old pulsars. A few more HAWC-detected sources have been suggested to be TeV halo candidates. In this paper, we search for possible GeV counterparts of three TeV halo candidates with Fermi Large Area Telescopes. We detect a new spatially extended GeV source in the vicinity of the TeV halo candidate 2HWC J1912+099, which is also detected by the High Energy Stereoscopic System (H.E.S.S.) (HESS J1912+101). We find that the size of the GeV source is significantly larger than that of the TeV emission measured by H.E.S.S., and a spatial template characteristic of a PWN can fit the GeV data. We suggest that the GeV source is an extended PWN powered by the central middle-aged pulsar PSR J1913+1011. This discovery favors the TeV halo scenario for the TeV source 2HWC J1912+099 (HESS J1912+101), although the possible shell-like morphology measured by H.E.S.S. challenges this interpretation. Alternatively, the TeV emission could be dominated by a supernova remnant via the hadronic process. Future multi-wavelength observations of the source and more precise measurements of the spatial profile of the TeV emission will be useful to distinguish between the two scenarios.